Game Theory

Introduction

Game theory is concerned with the decision-making process in situations where

outcomes depend upon choices made by one or more players. The word "game" is not

used in the conventional sense but describes any situation involving positive or negative

outcomes determined by the players' choices and, in some cases, chance. In order for

game theory to apply, certain assumptions must be made. The first is that each player is

rational, acting in his self-interest. In addition, the players' choices determine the

outcome of the game, but each player has only partial control of the outcome.

Game theory is the mathematical analysis of a conflict of interest to find optimal choices

that will lead to desired outcome under given conditions. To put it simply, it's a study of

ways to win in a situation given the conditions of the situation. While seemingly trivial

in name, it is actually becoming a field of major interest in fields like economics,

sociology, and political and military sciences, where game theory can be used to predict

more important trends.

Though the title of originator is given to mathematician John von Neumann, the first to

explore this matter was a French mathematician named Borel. In the 1930s, Neumann

published a set of papers that outlined the tenets of game theory and thus made way for

the first simulations which considered mathematical probabilities. This was used by

strategists during the Second World War, and since then has earned game theory a place

in the context of Social Science.

It may at first seem arcane to involve mathematics in something that seems purely based

on skill and chance, but game theory is in actuality a complex part of many branches of

mathematics including set theory, probability and statistics, and plain algebra. This

results from the fact that games are dictated by a given set of rules that can be used to

outline a set of possible moves which can be ranked by desirability and effectiveness,

and with information available, such a set can also be constructed for the opponent, thus

allowing predictions about the possible outcomes within a certain number of moves with

a probabilistic accuracy.

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Game Theory

Von Neumann and the development

of Game Theory

Emile Borel: The Forgotten Father of Game Theory?

In 1921, Emile Borel, a French mathematician, published

several papers on the theory of games. He used poker as

an example and addressed the problem of bluffing and

second-guessing the opponent in a game of imperfect

information. Borel envisioned game theory as being used

in economic and military applications. Borel's ultimate

goal was to determine whether a "best" strategy for a

given game exists and to find that strategy.

While Borel could be arguably called as the first mathematician to envision an organized

system for playing games, he did not develop his ideas very far. For that reason, most

historians give the credit for developing and popularizing game theory to John Von

Neumann, who published his first paper on game theory in 1928, seven years after Borel.

John Von Neumann

Born in Budapest, Hungary, in 1903, Von Neumann distinguished himself from his peers

in childhood for having a photographic memory, being able to memorize and recite back

a page out of a phone book in a few minutes. Science, history, and psychology were

among his many interests; he succeeded in every academic subject in school.

He published his first mathematical paper in collaboration with his tutor at the age of

eighteen, and resolved to study mathematics in college. He enrolled in the University of

Budapest in 1921, and over the next few years attended the University of Berlin and the

Swiss Federal Institute of Technology in Zurich as well. By 1926, he received his Ph.D.

in mathematics with minors in physics and chemistry.

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Game Theory

By his mid-twenties, von Neumann was known as a young mathematical genius and his

fame had spread worldwide in the academic community. In 1929, he was offered a job at

Princeton. Upon marrying his fiancee, Mariette, Neumann moved to the U.S. (Agnostic

most of his life, Von Neumann accepted his wife's Catholic faith for the marriage,

though not taking it very seriously.)

In 1937, Mariette left Von Neumann for J. B. Kuper, a physicist. Within a year of his

divorce, Von Neumann began an affair with Klara Dan, his childhood sweetheart, who

was willing to leave her husband for him.

Von Neumann is commonly described as a practical joker and always the life of the

party. John and Klara held a party every week or so, creating a kind of salon at their

house. Von Neumann used his phenomenal memory to compile an immense library of

jokes which he used to liven up a conversation. Von Neumann loved games and toys,

which probably contributed in great part to his work in Game Theory.

Beginning in 1927, Von Neumann applied new mathematical methods to quantum

theory. His work was instrumental in subsequent "philosophical" interpretations of the

theory.

For Von Neumann, the inspiration for game theory was poker, a game he played

occasionally and not terribly well. Von Neumann realized

that poker was not guided by probability theory alone, as

an unfortunate player who would use only probability

theory would find out. Von Neumann wanted to formalize

the idea of "bluffing," a strategy that is meant to deceive

the other players and hide information from them.

In his 1928 article, "Theory of Parlor Games," Von Neumann first approached the

discussion of game theory, and proved the famous Minimax theorem. From the outset,

Von Neumann knew that game theory would prove invaluable to economists. He teamed

up with Oskar Morgenstern, an Austrian economist at Princeton, to develop his theory.

Their book, Theory of Games and Economic Behavior, revolutionized the field of

economics. Although the work itself was intended solely for economists, its applications

to psychology, sociology, politics, warfare, recreational games, and many other fields

soon became apparent. 3

Game Theory

Although Von Neumann appreciated Game Theory's applications to economics, he was

most interested in applying his methods to politics and warfare, perhaps stemming from

his favorite childhood game, Kriegspiel, a chess-like military simulation. He used his

methods to model the Cold War interaction between the U.S. and the USSR, viewing

them as two players in a zero-sum game.

From the very beginning of World War II, Von Neumann was confident of the Allies'

victory. He sketched out a mathematical model of the conflict from which he deduced

that the Allies would win, applying some of the methods of game theory to his

predictions.

In 1943, Von Neumann was invited to work on the Manhattan Project. Von Neumann

did crucial calculations on the implosion design of the atomic bomb, allowing for a more

efficient, and more deadly, weapon. Von Neumann's mathematical models were also

used to plan out the path the bombers carrying the bombs would take to minimize their

chances of being shot down. The mathematician helped select the location in Japan to

bomb. Among the potential targets he examined was Kyoto, Yokohama, and Kokura.

"Of all of Von Neumann's postwar work, his development of the digital computer looms

the largest today." After examining the Army's ENIAC during the war, Von Neumann

came up with ideas for a better computer, using his mathematical abilities to improve the

computer's logic design. Once the war had ended, the U.S. Navy and other sources

provided funds for Von Neumann's machine, which he claimed would be able to

accurately predict weather patterns.

Capable of 2,000 operations a second, the computer did not predict weather very well,

but became quite useful doing a set of calculations necessary for the design of the

hydrogen bomb. Von Neumann is also credited with coming up with the idea of basing

computer calculations on binary numbers, having programs stored in computer's memory

in coded form as opposed to punchcards, and several other crucial developments. Von

Neumann's wife, Klara, became one of the first computer programmers.

Von Neumann later helped design the SAGE computer system designed to detect a

Soviet nuclear attack

In 1948, Von Neumann became a consultant for the RAND Corporation. RAND

(Research ANd Development) was founded by defense contractors and the Air Force as a 4

Game Theory

"think tank" to "think about the unthinkable." Their main focus was exploring the

possibilities of nuclear war and the possible strategies for such a possibility.

Von Neumann was, at the time, a strong supporter of "preventive war." Confident even

during World War II that the Russian spy network had obtained many of the details of

the atom bomb design, Von Neumann knew that it was only a matter of time before the

Soviet Union became a nuclear power. He predicted that were Russia allowed to build a

nuclear arsenal, a war against the U.S. would be inevitable. He therefore recommended

that the U.S. launch a nuclear strike at Moscow, destroying its enemy and becoming a

dominant world power, so as to avoid a more destructive nuclear war later on. "With the

Russians it is not a question of whether but of when," he would say. An oft-quoted

remark of his is, "If you say why not bomb them tomorrow, I say why not today? If you

say today at 5 o'clock, I say why not one o'clock?"

Just a few years after "preventive war" was first advocated, it became an impossibility.

By 1953, the Soviets had 300-400 warheads, meaning that any nuclear strike would be

effectively retaliated.

In 1954, Von Neumann was appointed to the Atomic Energy Commission. A year later,

he was diagnosed with bone cancer. The disease resulted from the radiation Von

Neumann received as a witness to the atomic tests on Bikini atoll.

Von Neumann maintained a busy schedule throughout his sickness, even when he

became confined to a wheelchair. It has been claimed by some that the wheelchair-bound

mathematician was the inspiration for the character of Dr. Strangelove in the 1963 film

Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb.

Von Neumann's last public appearance was in February 1956, when President

Eisenhower presented him with the Medal of Freedom at the White House. In April, Von

Neumann checked into Walter Reed Hospital. He set up office in his room, and

constantly received visitors from the Air Force and the Secretary of Defense office, still

performing his duties as a consultant to many top political figures.

John von Neumann died on February 8, 1957.

His wife, Klara von Neumann, committed suicide six years later.

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Game Theory

Dr. Marina von Neumann Whitman, John's daughter from his first marriage, was invited

by President Nixon to become the first woman to serve on the council of economic

advisers.

Concepts in Game Theory

Game

A conflict in interest among n individuals or groups (players). There exists a set of rules

that define the terms of exchange of information and pieces, the conditions under which

the game begins, and the possible legal exchanges in particular conditions. The entirety

of the game is defined by all the moves to that point, leading to an outcome.

Move

The way in which the game progresses between states

through exchange of information and pieces. Moves are

defined by the rules of the game and can be made in

either alternating fashion, occur simultaneously for all

players, or continuously for a single player until he

reaches a certain state or declines to move further. Moves

may be choice or by chance. For example, choosing a card from a deck or rolling a die is

a chance move with known probabilities. On the other hand, asking for cards in

blackjack is a choice move.

Information

A state of perfect information is when all moves are known to all players in a game.

Games without chance elements like chess are games of perfect information, while

games with chance involved like blackjack are games of imperfect information.

Strategy

A strategy is the set of best choices for a player for an entire game. It is an overlying plan

that cannot be upset by occurrences in the game itself.

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Game Theory

Payoff

The payoff or outcome is the state of the game at it's conclusion. In games such as chess,

payoff is defined as win or a loss. In other situations the payoff may be material (i.e.

money) or a ranking as in a game with many players.

Extensive and Normal Form

Games can be characterized as extensive or normal. A in extensive form game is

characterized by a rules that dictate all possible moves in a state. It may indicate which

player can move at which times, the payoffs of each chance determination, and the

conditions of the final payoffs of the game to each player. Each player can be said to

have a set of preferred moves based on eventual goals and the attempt to gain the

maximum payoff, and the extensive form of a game lists all such preference patterns for

all players. Games involving some level of determination are examples of extensive form

games.

The normal form of a game is a game where computations can be carried out completely.

This stems from the fact that even the simplest extensive form game has an enormous

number of strategies, making preference lists are difficult to compute. More complicated

games such as chess have more possible strategies that there are molecules in the

universe. A normal form game already has a complete list of all possible combinations of

strategies and payoffs, thus removing the element of player choices. In short, in a normal

form game, the best move is always known.

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Game Theory

Types of Games

One-Person Games

A one-person games has no real conflict of interest. Only the interest of the player in

achieving a particular state of the game exists. Single-person games are not interesting

from a game-theory perspective because there is no adversary making conscious choices

that the player must deal with. However, they can be interesting from a probabilistic

point of view in terms of their internal complexity.

Zero-Sum Games

In a zero-sum game the sum of the total possible payoffs at the end is zero since the

amounts won or lost are equal. Von Neumann and Oskar Morgenstern demonstrated

mathematically that n-person non-zero-sum game can be reduced to an n + 1 zero-sum

game, and that such n + 1 person games can be generalized from the special case of the

two-person zero-sum game. Another important theorem by Von Neumann, the minimax

theorem, states certain aspects of the maximal and minimal strategies of are part of all

two-person zero-sum games. Thanks to these discoveries, such games are a major part of

game theory.

Two-Person Games

Two-person games are the largest category of familiar

games. A more complicated game derived from 2-person

games is the n-person game. These games are

extensively analyzed by game theorists. However, in

extending these theories to n-person games a difficulty

arises in predicting the interaction possible among

players since opportunities arise for cooperation and

collusion.

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Game Theory

Zero-Sum Games

A zero-sum game is one in which no wealth is created or destroyed. So, in a two-player

zero-sum game, whatever one player wins, the other loses. Therefore, the players share

no common interests. There are two general types of zero-sum games: those with perfect

information and those without.

In a game with perfect information, every player knows the results of all previous moves.

Such games include chess, tic-tac-toe, and Nim. In games of perfect information, there is

at least one "best" way to play for each player. This best strategy does not necessarily

allow him to win but will minimize his losses. For instance, in tic-tac-toe, there is a

strategy that will allow you to never lose, but there is no strategy that will allow you to

always win. Even though there is an optimal strategy, it is not always possible for players

to find it. For instance, chess is a zero-sum game with perfect information, but the

number of possible strategies is so large that it is not possible for our computers to

determine the best strategy.

In games with imperfect information, the players do not know all of the previous moves.

Often, this occurs because the players play simulataneously. Here are some examples of

such games:

Game 1

Suppose two people are playing a simple game with nickels and quarters. At the same

time, they each put out either a nickel or a quarter. If at least one player plays a nickel,

player 1 gets both coins. Otherwise, player 2 gets both. Naturally, both players wish to

gain as much money as possible. How should they play in order to do this?

Game 2

Suppose two people are playing a similar game with nickels and quarters. Now, if player

1 plays a nickel, player 2 gives him 5 cents. If player 2 plays a nickel and player 1 plays

a quarter, player 1 gets 25 cents. If both players play quarters, player 2 gets 25 cents.

Game 39

Game Theory

We still have two people playing a game with nickels and quarters. Now, if both players

play the same coin, player 2 gives player 1 the average value of the coins; otherwise,

player 1 gives player 2 the average value of the coins.

Although the three games seem similar, the methods used to find the best strategies in

each are very different. Game 1 is solved by eliminating dominant strategies, game 2's

solution is known as a saddle point, and game 3 requires a mixed strategy.

Game 1 – Dominant Strategies

Suppose two people are playing a simple game with nickels and quarters. At the same

time, they each put out either a nickel or a quarter. If at least one player plays a nickel,

player 1 gets both coins. Otherwise, player 2 gets both. Naturally, both players wish to

gain as much money as possible. How should they play in order to do this?

We can assign payoff matrices to such games that define the payoffs

that players will get based on the strategies they use. In this example,

each player has only two strategies--put out a nickel or put out a

quarter. Here is a payoff matrix for player 1:

Player 2

Nickel Quarter

Player 1Nickel 5 25

Quarter 5 -25

The rows represent player 1's possible strategies, and the columns represent player 2's

possible strategies. If player 1 and player 2 both play nickels (the top left entry), player 1

wins player 2's nickel so gains 5 cents. On the other hand, if both play quarters (the

bottom right entry), player 2 wins player 1's quarter, so player 1 loses 25 cents.

Notice that every entry in the first row is greater than all of the entries

in that column. In other words, playing a nickel is always at least as

good as playing a quarter for player 1. So, playing a nickel is called a

dominant strategy, and it dominates the strategy of playing a quarter.

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Game Theory

It is never advantageous to play a dominated strategy, so we can

reduce our payoff matrix to reflect this:

Player 2

Nickel Quarter

Player 1Nickel 5 25

Now, the nickel strategy for player 2 also dominates. So, playing nickels is the best

strategy for both players. Notice that, if either plays quarters, he will not gain more

money than if he had just played nickels.

Game 2 – Saddle Points

This game differs from game 1 in that it has no dominant strategies. The rules are as

follows: If player 1 plays a nickel, player 2 gives him 5 cents. If player 2 plays a nickel

and player 1 plays a quarter, player 1 gets 25 cents. If both players play quarters, player 2

gets 25 cents. We get a payoff matrix for this game:

Player 2

Nickel Quarter

Player 1Nickel 5 25

Quarter 25 -25

Notice that there are no longer any dominant strategies. To solve this game, we need a

more sophisticated approach. First, we can define lower and upper values of a game.

These specify the least and most (on average) that a player can expect to win in the game

if both player play rationally. To find the lower value of the game, first look at the

minimum of the entries in each row. In our example, the first row has minimum value 5

and the second has minimum -25. The lower value of the game is the maximum of these

numbers, or 5. In other words, player 1 expects to win at least an average of 5 cents per

game. To find the upper value of the game, do the opposite. Look at the maximum of

every column. In this case, these values are 25 and 5. The upper value of the game is the

minimum of these numbers, or 5. So, on average, player 1 should win at most 5 cents per

game.

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Game Theory

Nickel Quarter Min

Nickel 5 5 5

Quarter 25 -25 -25

Max 25 5

Notice that, in our example, the upper and lower values of the game are the same. This is

not always true; however, when it is, we just call this number the pure value of the game.

The row with value 5 and the column with value 5 intersect in the top right entry of the

payoff matrix. This entry is called the saddle point or minimax of the game and is both

the smallest in its row and the largest in its column. The row and column that the saddle

point belongs to are the best strategies for the players. So, in this example, player 1

should always play a nickel while player 2 should always play a quarter.

Game 3 – Mixed Strategies

The rules of game 3 were as follows: two players have nickels and quarters. At the same

time, they each play one coin. If both players play the same coin, player 2 gives player 1

the average value of the coins; otherwise, player 1 gives player 2 the average value of the

coins. Here is the payoff matrix for this game:

Player 2

Nickel Quarter

Player 1Nickel 5 -15

Quarter -15 25

The lower value of this game is -15 while the upper value is 5. Can we find a pure value

for the game? According to the Minimax Theorem, one of the most important results in

game theory, we can. The Minimax Theorem states that every finite, two-person, zero-

sum game has a value V that is the average amount that one player can expect to win if

both players act sensibly.

Suppose player 2 knows which coin player 1 will play on each turn. Then it will be easy

for player 2 to play a coin that makes player 2 lose money. Therefore, player 1 can't play

with a pattern. Instead, he must use a mixed strategy, in which he randomly chooses to

play a nickel or quarter on each turn. However, it is not necessarily true that he should

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Game Theory

play each strategy half the time. He may want to weight the strategies differently,

playing one with probability p and the other with probability 1 - p. How do we figure out

p?

It turns out that one property of the value of a game is that, if player 1 plays his optimal

strategy, he will achieve exactly the value of the game no matter what the other player

does (as long as the other player has no dominant strategies). In particular, the yield

when player 1 plays agains player 2's two different pure strategies should be the same. In

other words, if player 1 uses his optimal strategy, he will get the same amount of money

whether player 2 always plays nickels or always plays quarters. Let's suppose that player

2 always plays nickels. Player 1 plays nickels p of the time so gains 5 cents p of the time.

The other 1 - p of the time, he loses 15 cents. Overall, he wins 5p - 15(1 - p) = 20p - 15.

Now, suppose player 2 always plays quarters. Player 1 plays nickels p of the time so

loses 15 cents p of the time. The rest of the time, he wins 25 cents. Overall, he wins -15p

+ 25(1 - p) = 25 - 40p. Because he should win the same in both situations, the two

winnings are the same. So, 20p - 15 = 25 - 40p. Solving for p, we find that it is 2/3. To

find the amount that player 1 expects to win, we just plug this back into either of the

equations and find that he should win an average of -5/3 per game. Even if player 2

figures out this strategy, he cannot do anything to change it.

Similarly, we can look at the payoff matrix from player 2's point of view and find a

mixed strategy for player 2. If we do so, we find that player 2 should play nickels 2/3 of

the time and quarters 1/3 of the time. If he does so, he should win an average of 5/3 cents

per game.

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Game Theory

Strategies of Play

The Minimax algorithm is the most well-known strategy of play of two-player, zero-sum

games. The minimax theorem was proven by John von Neumann in 1928. Minimax is a

strategy of always minimizing the maximum possible loss which can result from a choice

that a player makes. Before we examine minimax, though, let's look at some of the other

possible algorithms.

Maximax

Maximax principle counsels the player to choose the strategy that yields the best of the

best possible outcomes. For example, let's consider a zero-sum game where two players

simultaneously put either a blue or a red card on the table. If player 1 puts a red card

down on the table, whichever card player 2 puts down, no one wins anything. If player 1

puts a blue card on the table and player 2 puts a red card, then player 2 wins $1,000 from

player 1. Finally, if player 1 puts a blue card on the table and player 2 puts a blue card

down, then player 1 wins $1,000 from player 2.

The payoff matrix for player 1 is shown in this table:

Player 2

Blue Red

Player 1Blue 1,000 -1,000

Red 0 0

Going by maximax principle, player 1 will always play the blue card, since it has the

maximum possible payoff of 1,000. But as can be clearly seen from the table, assuming

player 2 is rational, he will never play the blue card, since the red card gives him the

dominant strategy. In such a case, if player 1 plays by the maximax rule, he will always

lose.

The maximax principle is inherently irrational, as it does not take into account the other

player's possible choices. Maximax is often adopted by naive decision-makers such as

young children.

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Game Theory

Maximin

Maximin is solely a one-person game strategy, i.e. a principle which may be used when a

person's "competition" is nature, or chance. Whereas the maximax principle is ultra-

optimistic, expecting the best possible payoff, the maximin is ultra-pessimistic, expecting

the worst possible payoff. It involves choosing the best of the worst possible outcomes.

A simple example of a slot machine game may be used. A player has only two choices to

make -- to gamble or not to gamble. If he gambles, he risks losing his bet (say, $1), but

also has a chance to win the maximum payoff (say, $10,000). If he does not gamble, he

can neither win nor lose.

The payoff matrix looks like this:

Chance

Win Lose

PlayerGamble

1000

0-1

Not Gamble 0 0

According to the maximin principle, the player should never gamble, because he faces a

risk of losing $1. It is clear that the maximin principle is quite inefficient, since it

discourages taking any risks, no matter how high the reward may be.

Minimax for One-Person Games

The Minimax Regret Principle is based on the Minimax Theorem advanced by John von

Neumann, but is geared only towards one-person games. It relies on the concept of regret

matrices. To demonstrate, consider an example of a company trying to decide whether or

not it should support a research project. Let's assume that the research project costs c

units. If it succeeds, it will bring in a return of r units. If it fails, it will obviously not

bring in anything.

The payoff matrix for the company looks like this:

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Game Theory

Research

Succeeds Fails

CompanySupports Research R - C - C

Neglect Research 0 0

By the maximax principle, a company should always support research if the expected

return on it is greater than its cost. By maximin, the company should never support

research, since it is risking the cost of the research. Minimax is slightly more

complicated.

We need to come up with a matrix that shows the "opportunity cost," or regret, of the

player, depending on each possible decision. For example, if the company supports the

research and it fails, the company's regret will be c, the price of research. If the company

supports the research and it succeeds, the company will have no regrets. If the company

neglects research and it would have succeeded, its regret value is r-c, the return on the

research. So, the minimax regret matrix will look like this:

Research

Succeeds Fails

CompanySupports Research 0 C

Neglect Research R - C 0

The object is to minimize the maximum possible regret. It is not obvious from the above

matrix what the maximum value is. That is, is c greater than r-c? If (r-c) > c, the

company should support research. If (r-c) < c, the company should not. In other words,

the company should support research if c < r/2, or, if the expected return on research is

more than twice its cost.

Minimax for Two-Person Games

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Game Theory

In a two-person, zero-sum game, a person can win only if the other player loses. No

cooperation is possible. Andrew Colman's Game Theory and Experimental Games shows

the following historical example:

In 1943, the Allied forces received reports that a Japanese convoy would be heading by

sea to reinforce their troops. The convoy could take on of two routes – the Northern or

the Southern route. The Allies had to decide where to disperse their reconnaissance

aircraft - in the north or the south - in order to spot the convoy as early as possible. The

following payoff matrix shows the possible decisions made by the Japanese and the

Allies, with the outcomes expressed in the number of days of bombing the Allies could

achieve with each possibility:

Japanese Route

North South

Allies ReconnaissanceNorth 2 2

South 1 3

By this matrix, if the Japanese chose to take the southern route while the Allies decided

to focus their recon planes in the north, the convoy would be bombed for 2 days. The

best outcome for the Allies would be if they placed their airplanes in the south and the

Japanese took the southern route. The best outcome for the Japanese would be to take the

northern route, provided the Allies were looking for them in the south.

To minimize the worst possible outcome, the Allies would have to choose the north as

the focus of their reconnaisance efforts. This ensures them 2 days of bombing, whereas

they risk only 1 day of bombing if they focus on the south. Therefore, by minimax, the

best strategy for them would be to focus on the north.

The Japanese can use the same strategy. The worst possible outcome for them is the 3

days of bombing which might occur if they took the southern route. Therefore, the

Japanese would take the northern route.

It is, in fact, what had occurred: both the Allies and the Japanese chose the north, and the

Japanese convoy was bombed for 2 days.

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Game Theory

The previous matrix had a saddle point, meaning that both the Japanese and the Allies

settled on the (North, North) square as the best outcome for both of them. Neither could

do any better if the opponent was rational. In this case, the maximin and the minimax

strategies produce the same result. Notice that if the Allies were following maximax,

they would choose the South, and surely forfeit one day of bombing.

Mixed Strategies and Randomization

In some cases, there is no saddle point, and the players have to choose their strategies

with a degree of randomness, as in the following simple game, called "Matching

Pennies." Two players simultaneously place a penny on a table, either heads up or tails

up. If the pennies are facing the same way, player 1 gets to keep both pennies.

Otherwise, player 2 gets to keep both. The payoff matrix for player 1 looks like this:

Player 2

Heads Tails

Player 1Heads 1 -1

Tails -1 1

There is no clearly defined strategy for each player. The best way to play is to choose the

position of the coin randomly. If either player follows this strategy, then in the long run,

the payoffs for each will be 0. Notice that if, say, player 1 uses a 50/50 strategy, while

player 2 plays heads 75% of the time, in the long run, both players will still have payoffs

of 0. But if player 2 follows the 75/25 strategy, then player 1 can easily take advantage of

it by playing heads more frequently, and therefore winning more frequently. So, it is

important for each player to not only maintain a random strategy, but to also analyze the

strategy of the other player.

Applications to Computing

In computer simulations for cases such as this, it is important not to program the

computer with a specific strategy in advance, but let it decide it at run-time. If the

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Game Theory

computer does not maintain unpredictability, then the opposing player may use this

knowledge to his advantage. Many computer games suffer because although the

computer is programmed with a strong strategy, it becomes predictable and easy to take

advantage of.

On the other hand, the computer might very well benefit if it recognizes a predictable

strategy on the part of an opponent. Even in such a simple game as "Matching Pennies,"

where a 50/50 is called for, while the computer may follow a 50% algorithm for deciding

whether to play heads or tails, a human cannot come up with completely random

numbers. In fact, it has been observed that humans tend to play heads slightly more

often. If a computer recognizes that the probability of its opponent of picking heads is

slightly higher, it may adjust its own strategy to have an advantage.

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Game Theory

Non-Zero-Sum Games

The theory of zero-sum games is vastly different from that of non-zero-sum games

because an optimal solution can always be found. However, this hardly represents the

conflicts faced in the everyday world. Problems in the real world do not usually have

straightforward results. The branch of Game Theory that better represents the dynamics

of the world we live in is called the theory of non-zero-sum games. Non-zero-sum games

differ from zero-sum games in that there is no universally accepted solution. That is,

there is no single optimal strategy that is preferable to all others, nor is there a

predictable outcome. Non-zero-sum games are also non-strictly competitive, as opposed

to the completely competitive zero-sum games, because such games generally have both

competitive and cooperative elements. Players engaged in a non-zero sum conflict have

some complementary interests and some interests that are completely opposed.

A Typical Example

The Battle of the Sexes is a simple example of a typical non-zero-sum game. In this

example a man and his wife want to go out for the evening. They have decided to go

either to a ballet or to a boxing match. Both prefer to go together rather than going alone.

While the man prefers to go to the boxing match, he would prefer to go with his wife to

the ballet rather than go to the fight alone. Similarly, the wife would prefer to go to the

ballet, but she too would rather go to the fight with her husband than go to the ballet

alone. The matrix representing the game is given below:

Husband

Boxing Match Ballet

WifeBoxing Match 2, 3 1, 1

Ballet 1, 1 3, 2

The wife's payoff matrix is represented by the first element of the ordered pair while the

husband's payoff matrix is represented by the second of the ordered pair.

From the matrix above, it can be seen that the situation represents a non-zero-sum, non-

strictly competitive conflict. The common interest between the husband and wife is that

20

Game Theory

they would both prefer to be together than to go to the events separately. However, the

opposing interests is that the wife prefers to go to the ballet while her husband prefers to

go to the boxing match.

Analyzing a Non-Zero-Sum Game

Communication

It is conventional belief that the ability to communicate could never work to a player's

disadvantage since a player can always refuse to exercise his right to communicate.

However, refusing to communicate is different from an inability to communicate. The

inability to communicate may work to a player's advantage in many cases.

An experiment performed by Luce and Raiffa compares what happens when player can

communicate and when players cannot communicate. Luce and Raiffa devised the

following game:

A B

A 1, 2 3,1

B 0, -200 2, -300

If Susan and Bob cannot communicate, then there is no possibility of threats being made.

So, Susan can do no better than to play strategy A and Bob can do no better than to play

strategy a. Susan, therefore gains 1 and Bob gains 2. However, when communication is

allowed, the situation is complicated. Susan can threaten Bob by saying that she will play

strategy B unless Bob commits himself to playing strategy b. If Bob submits, Susan will

gain 2 and Bob will lose 1 (as opposed to Susan gaining 1 and Bob gaining 2 when

communication is not allowed).

Restricting Alternatives

The Battle of the Sexes example given above seems to be an unsolvable dilemma.

However, this problem can be solved it either the husband or the wife resticts the others'

alternatives. For example, if the wife buys two tickets for the ballet, indicating that she is

definitely not going to the boxing match, the husband would have to go to the ballet

along with his wife in order to maximize his self-interest. Because the wife bought the

two tickets, the husbands optimal payoff, now, would be to go along with his wife. If he

were to go to the boxing match alone, he would not be maximizing his self-interest. 21

Game Theory

The Number of Times the Game is "Played"

If the game is played only once, players do not have to fear retaliation from their

opponents, so they may play differently than they would in a game played repeatedly.

Typical Non-Zero-Sum Games:

Prisoner's Dilemma

Chicken and Volunteer's Dilemma

Deadlock and Stag Hunt

The Prisoner’s Dilemma

The Prisoner's Dilemma game was first proposed by Merrill Flood in 1951. It was

formalized and defined by Albert W. Tucker. The name refers to the following

hypothetical situation:

Two criminals are captured by the police. The police suspect that they are responsible for

a murder, but do not have enough evidence to prove it in court, though they are able to

convict them of a lesser charge (carrying a concealed

weapon, for example). The prisoners are put in separate

cells with no way to communicate with one another and

each is offered to confess.

If neither prisoner confesses, both will be convicted of the lesser offense and sentenced

to a year in prison. If both confess to murder, both will be sentenced to 5 years. If,

however, one prisoner confesses while the other does not, then the prisoner who

confessed will be granted immunity while the prisoner who did not confess will go to jail

for 20 years.

What should each prisoner do?

Discussion of Prisoners Dilemma

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Game Theory

To help us determine the answer, let's come up with a payoff matrix for each prisoner.

The value in each cell is the time spent in prison, so the prisoners will try to end up in the

matrix cell with the lowest number. The first number of each pair refers to the prison

time of prisoner 1, and the second number to prisoner 2.

Prisoner 2

Confess Not Confess

Prisoner 1Confess 5, 5 0, 20

Not Confess 20, 0 1, 1

Let's assume the role of prisoner 1. We're looking to minimize our prison time. Since we

have no way of knowing whether our partner in crime has confessed, let's first assume

that he has not. If Prisoner 1 doesn't confess either, both will go to prison for 1 year. Not

bad. But, if Prisoner 1 confesses, he will go free, while his partner rots away in jail. We'll

assume that there is no "honor among thieves" and each prisoner only cares about

minimizing his jail time. From the above discussion, it is obvious that if Prisoner 2 does

not confess, Prisoner 1 is definitely better off confessing.

Now let's look at the other possibility. Say prisoner 2 confesses. If Prisoner 1 does not

confess, he will go to jail for 20 years. But if he does confess, he will get only 5 years in

prison. It is clearly better to confess in this case as well.

So is that it? Is the problem solved? Is each prisoner better off confessing? Well, it may

seem so from the above discussion, but if we look at the payoff matrix, it is clear that the

best payoff for both prisoners is when neither confesses! But game theory advocates that

both confess.

This "game" can be generalized to any situation when two players are in a non-

cooperative situation where the best all-around situation is for both to cooperate, but the

worst individual outcome is to be cooperating player while the other player defects.

On the one hand, it is tempting to defect, or confess. Since you have no way of

influencing the other player's decision, no matter what he does, you're better off

confessing. But on the other hand, you're both in the same boat. Both of you should be

sensible enough to realize that cheating undermines the common good.23

Game Theory

There is no single "right" solution to the Prisoner's Dilemma (that's why it's a dilemma).

Its implications carry into psychology, economics, and many other fields.

The Flood Dresher Experiment

Many experiments have been done on the Prisoner's Dilemma, to try to gauge the normal

human behavior in a prisoner's dilemma-type situation.

The Flood-Dresher Experiment was a prisoner's dilemma game run 100 times between 2

players. In this case, the game was unfair - one of the players had an inherent advantage

over the other player, but the payoff matrix layout was still a prisoner's dilemma. The

following is the table used in the experiment. (In this case, the payoffs are positive, that

is, each rational player seeks to maximize the value in the matrix cell he ends up in.)

Prisoner 2

Defect Cooperate

Prisoner 1Cooperate -1, 2 0.5, 1

Defect 0, 0.5 1, -1

As in the Prisoner's Dilemma, both players are better off defecting. But when both

defect, they do relatively poorly. On the other hand, if both choose their "worse" strategy

consistently, they should both gain.

In the 100 trials, Player 1 chose to cooperate 68 times, and Player 2 78 times. Player 1

began the game expecting both players to defect. Player 2 realized the value of

cooperation and started cooperating. Both players started cooperating after the first 10 or

so moves, though Player 1 would defect on a regular basis, unhappy that his payoff

wasn't as big as Player 2's. This in turn brought retaliation from Player 2, who would

defect on the next move.

Each player kept a log of comments for each move. Some of those comments are quite

amusing. "The stinker," writes Player 2 after Player 1's defection. "He's a shady

character... A shiftless individual--opportunist, knave... He can't stand success."

24

Game Theory

The players' comments reflect their concern about the final few moves. Both seem to

realize that it would make sense for both to defect on move 100, since no retaliation from

the other player is possible. Player 1 worries about starting to defect earlier than Player 2

so that he has the advantage. As the game was played, both players cooperated on moves

83 through 98. On move 99, Player 1 defected, and on move 100, both defected.

It is clear that the long-term prospect of the game encouraged cooperation. Since the

game was played multiple times, it became beneficial for both players to cooperate. On

move 100, however, the game suddenly becomes a regular prisoner's dilemma, and both

players defect, as game theory advocates they should (although if they both cooperated

they would ensure themselves a gain of 0.5 points).

This reasoning is troubling though. Since both players must realize that they will both

defect on move 100, move 100 does not have to figure into the game. It can then be

thought that move 99 is really the last move in the game, since both players are

obviously going to defect on move 100. But if move 99 is the last move, both players

should defect, since no retaliation is possible (both players will defect anyway on move

100, no matter what the other player did on move 99). So both players should defect on

move 99 as well. Then, move 98 can be thought of as the last move in the game. This

line of reasoning can be extended indefinitely until move 1. So should both players

always defect?

Clearly not, since if they both cooperate, they will gain more than if both defect.

This paradox is still unresolved. William Poundstone, in Prisoner's Dilemma, says that

"Both Flood and Dresher say they initially hoped that someone would 'resolve' the

prisoner's dilemma. They expected someone to come up with a new and better theory of

non-zero-sum games. The solution never came. The prisoner's dilemma remains a

negative result - A demonstration of what's wrong with theory, and indeed, the world."

Axelrod’s Tournament

In 1980, Robert Axelrod, professor of political science at the University of Michigan,

held a tournament of various strategies for the prisoner's dilemma. He invited a number

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Game Theory

of well-known game theorists to submit strategies to be run by computers. In the

tournament, programs played games against each other and themselves repeatedly. Each

strategy specified whether to cooperate or defect based on the previous moves of both

the strategy and its opponent.

Some of the strategies submitted were:

Always defect: This strategy defects on every turn. This is what game

theory advocates. It is the safest strategy since it cannot be taken advantage of.

However, it misses the chance to gain larger payoffs by cooperating with an

opponent who is ready to cooperate.

Always cooperate: This strategy does very well when matched against

itself. However, if the opponent chooses to defect, then this strategy will do

badly.

Random: The strategy cooperates 50% of the time.

All of these strategies are prescribed in advance. Therefore, they cannot take advantage

of knowing the opponent's previous moves and figuring out its strategy.

The winner of Axelrod's tournament was the TIT FOR TAT strategy. The strategy

cooperates on the first move, and then does whatever its opponent has done on the

previous move. Thus, when matched against the all-defect strategy, TIT FOR TAT

strategy always defects after the first move. When matched against the all-cooperate

strategy, TIT FOR TAT always cooperates. This strategy has the benefit of both

cooperating with a friendly opponent, getting the full benefits of cooperation, and of

defecting when matched against an opponent who defects. When matched against itself,

the TIT FOR TAT strategy always cooperates.

Several variations to TIT FOR TAT have been proposed. TIT FOR TWO TATS is a

forgiving strategy that defects only when the opponent has defected twice in a row. TWO

TITS FOR TAT, on the other hand, is a strategy that punishes every defection with two

of its own.

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Game Theory

TIT FOR TAT relies on the assumption that its opponent is trying to maximize his score.

When paired with a mindless strategy like RANDOM, TIT FOR TAT sinks to its

opponent's level. For that reason, TIT FOR TAT cannot be called a "best" strategy.

It must be realized that there really is no "best" strategy for prisoner's dilemma. Each

individual strategy will work best when matched against a "worse" strategy. In order to

win, a player must figure out his opponent's strategy and then pick a strategy that is best

suited for the situation.

Multi-Person Prisoner’s Dilemma

The n-person prisoner's dilemma (NPD) is basically the Prisoner's Dilemma with more

than two players. The NPD emerged during the early 1970's and quickly became popular

among social theorists and economists. The sudden interest in NPD occurred mainly

because of the economic and social developments during the late 60s and early 70s. At

this time, problems such as inflation, voluntary wage restraint, the energy crisis, and

environmental pollution were pressing issues. This era of history, however, is better

known for the increasing international tension between the U.S. and the Soviet Union.

Both superpowers were engaged in mass production of nuclear weapons, creating a very

real threat to the existence of the entire world. With the proliferation of nuclear weapons

came the issue of multilateral disarmament. The various social, political, and economic

tensions of the 70's can all be modeled by the NPD, indicating the remarkable range of

real-world problems that NPDs can simulate.

Many real-world problems, be they social, political, or economic, can be modeled as an

NPD. In economics, an interesting example concerns the "invisible hand theory" and

how it applies to the labor market.

In 1776, economist Adam Smith introduced the theory of the "invisible hand" which still

remains the cornerstone of traditional economics. The "invisible hand", in short, is what

dictates the motion of the economy. It is not a single individual or government that

controls its motion, but is instead motivated by every person who participates in the

economy.

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Game Theory

In the labor market, companies hiring workers are consumers and those looking for jobs

are the producers. That is, job seekers have a product to sell, namely their skills and the

companies want to buy their labor. The "invisible hand" which dictates the labor market

decides the wages that companies will pay.

An example of how and NPD can be used to model the labor market is as follows: Every

trade union's individual self-interest is to negotiate wages that exceed the rate of inflation

in the economy as a whole. However, if all trade unions negotiated wages to benefit their

own self-interest, the prices of goods and services go up and everyone is worse off than

if they had all exercised restraint.

In order to solve this problem, the British Labour Party issued a Manifesto (1974) which

contained an outline of a "social contract" whose aim was to encourage trade unions to

exercise voluntary wage restraint in order to decrease the rate of inflation. The "social

contract" was designed to encourage collective rationality in wage bargaining over

individual rationality. However, this solution was unsuccessful because it did not change

the underlying strategic structure of the wage bargaining game.

Another type of NPD that is readily evident in the real

world is that which simulates situations where

resources are scarce. For example, when there is a

shortage of any resource, such as water or energy,

there is usually a call for conservation. However, and

individual only benefits from restraint if everyone else restrains as well. However,

restraint of an individual is unnecessary. That is, if everyone else restrains then it would

make much of a difference if you didn't restrain. On the other-hand, if you restrain and

no one else does, then your attempt at conservation is futile. Therefore, it is everyone's

individual self-interest to NOT conserve. However, if everyone acts individualistically,

all are worse off.

One last interesting example of an NPD is called the ‘tragedy of the commons’. Suppose

there are six farmers who each owns one cow that weighs 1000 lbs. These six farmers

share one plot of grazing land, a plot that can maximally sustain six cows without

deterioration from overgrazing. For every additional cow that is added, the weight of

every animal decreases by 100 lbs. Suppose every farmer had the opportunity to add one 28

Game Theory

cow. If one farmer decides to add one cow, then his wealth increases since he will now

have two cows that weigh 900 lbs each instead of just one cow that weighs 1000 lbs.

Each of the six farmers, if they act in their own self-interest, will also add another cow.

However, if all six farmers do add another cow, then each farmer ends up worse off. That

is, each farmer will have two cows that weighs 400 lbs each instead of one cow that

weighs 1000 lbs.

Small farmers in England during the period of the enclosures in the eighteenth century

became impoverished because of this NPD situation.

All multi-person prisoner's dilemmas share a common underlying strategic structure.

Therefore, any game that satisfies the following criteria is an NPD by definition:

each player has two options: cooperate or defect

defecting is the dominant strategy for each player (i.e. each player is better off

choosing to defect than to cooperate no matter how many other players choose to

cooperate)

the dominant strategies (to defect) intersect at a deficient equilibrium point (if all

players choose to defect, the outcome is worse than if each player had chosen

non-dominant strategies (to cooperate))

Chicken

There is a game called Chicken, in which two people drive two very fast cars towards

each other from opposite ends of a long straight road. If one of them swerves before the

other, he is called a chicken. Of course, if neither swerves, they will crash. The worst

possible payoff is to crash into each other, so we assign this a value 0. The best payoff is

to have your opponent be the chicken, so we assign this a value 3. The next to worst

possibility is to be the chicken, so we assign this a value 1. The last possibility is that

both drivers swerve. Then, neither has less honor than the other, so this is preferable to

being the chicken. However, it is not quite as good as being the victor, so we assign it a

value 2. We could assign a payoff matrix to this:

Swerve Drive Straight

Swerve 2, 2 1, 3

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Game Theory

Drive Straight 3, 1 0, 0

Unlike the prisoner's dilemma, mutual defection is the worst outcome in chicken. Both

players want to do the opposite of what the other player does.

Volunteers’ Dilemma

The version of chicken with more than two players is

known as the volunteer's dilemma. In a volunteer's

dilemma, one player needs to take an action that will

benefit all of the players. For instance, suppose James

Bond, Paris Carver, and Wai Lin are locked in three

sound-proof cells by Elliot Carver. In one hour, Elliott

will release poison gas into their cells unless at least one of the three pushes a button.

Whoever pushes the button will be immediately killed, but the other two will be released

immediately. The three cannot communicate or coordinate their efforts.

If any of the three are to survive, one of them must sacrifice himself or herself. The least

disturbing case is when all three reach the same conclusion about who should be

sacrificed. In this case, the martyr will push the button, and the others will be spared. A

second possibility is that all players decide to save each other. Then there will be a race

to push the button first. The most disturbing case is when each player decides that he or

she should be saved. When this happens, none push the button and the clock ticks away.

Suppose that you are Bond, and Paris and Wai Lin have not pushed the button by the end

of 59 minutes. It seems that they have decided that you should sacrifice yourself, but you

don't want to do that. There is no point in vowing to never push the button because then

all three of you will die. Ideally, you want to push the button in the last possible second.

However, there is no way to determine exactly when this is, so the resolution of the

conflict rides on chance and reflexes.

Other Dilemmas

These dilemmas are examples of games in which both players share the same

preferences. These games are known as symmetric games. In these games, neither player

has a privileged position. In this sense, they can often model the real world.

30

Game Theory

Deadlock

The payoff matrix for deadlock looks something like this:

Cooperate Defect

Cooperate 1, 1 0, 3

Defect 3, 0 2, 2

Each player does better defecting no matter what his partner does. Unlike the prisoner's

dilemma though, it is better for them to both defect than to both cooperate. This is called

deadlock because the two players will decide not to cooperate. This situation sometimes

arises when two countries do not want to disarm so fail to reach arms control

agreements.

Stag Hunt

The philosopher Jean-Jacques Rousseau imagined a situation like this:

In early societies, people formed alliances to hunt deer. If even one person in the group

did not help with the hunt, the deer would be lost. The hunters were sometimes tempted

to leave the hunt by seeing rabbits, but they preferred deer to rabbit. However, only one

person was needed to catch a rabbit. From a game theory perspective, the best strategy is

to hunt the deer, but people may decide to hunt the rabbit because they believe others

may defect from the hunt also.

Countries face the same dilemma in situations involving nuclear weapons. Each country

generally believes that the world would be better if no countries possessed nuclear

weapons. However, the temptation to build up a nuclear arsenal arises because each

country is afraid that other countries may stash nuclear warheads and undermine

international security.

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Game Theory

Applications of Game Theory

Though at first glance the idea of game theory sounds trivial, applications of game theory

are extensive. Von Neumann and Morgenstern originally applied their models of games

to economic analysis. Each factor in the market, such as seasonal preferences, buyer

choice, changes in supply and material costs, and other such market factors can be used

to describe strategies to maximize the outcome and thus the profit. However, game

theory can be also used to simply study economics of the past and interactions of

different factors in a matter. It can also be used to investigate matters such as monetary

distributions and their effects on other outcomes.

Military strategists have turned to game theory to play "war games." Usually, such

games are not zero-sum games, for loses to one side are not won by the other, and they

have been criticized as potentially dangerous oversimplification of necessarily factors.

Economic situations are also more complicated than zero-sum games, but those factors

only require readjustments to the strategy over time. Sociologists have taken an interest

in game theory, and have developed an entire branch dedicated to group decision

making. Immunization procedures and vaccine or other medication tests are analyzed by

epidemiologists using game theory.

The properties of n-person non-zero-sum games can be used to study different aspects of

social sciences as well. Matters such as distribution of power, interactions between

nations, the distribution of classes and their effects of government, and many other

matters can be easily investigated by breaking the problem down into smaller games,

each of whose outcomes affect the final result of a larger game.

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Game Theory

Philosophy

Philosophers are increasingly becoming interested in Game Theory because it provides a

way of elucidating the logical difficulty of philosophers such as Hobbes, Rousseau, Kant

and other social and political theorists.

Rationality and the pursuit of self-interest:

According to Bertrand Russell “‘Reason’ has a perfectly clear and precise meaning. It

signifies the choice of the right means to an end that you wish to achieve". This is the

interpretation of 'reason' that most contemporary philosophers favor. However, many

philosophers have pointed out situations where the concept of rationality seems to break

down. The situations are those who strategic structures resemble that of the Prisoner's

Dilemma.

An example of a multiple person Prisoner's Dilemma is as follows: Suppose that during a

drought, a person must decide whether he should act in his own self-interest and water

the garden or whether he should exercise restraint and conserve water. No matter what

the other community members do, a person is always better off watering his garden

because this is the right means to the end that he desires. The reasoning for this is that it

is unnecessary for one person to exercise restraint if the most other community members

are restraining as well. Even if the rest of the community doesn't exercise restraint, it is

futile for just one person to do so since one person does not have that big of an impact on

the whole water supply.

The paradox is that if the entire community reasons this way, the water supply will dry

up completely but if each community member cooperates and exercises restraint (acts

irrationally) the water supply will be spared. Moral philosopher, Derek Parfit, believes

that cooperation, instead of being the irrational choice, can be a rational course of action.

Parfit has proposed several solutions to the Prisoner's dilemma so that cooperation

becomes the reasonable choice. One solution involves changing the entire structure of

the game so that it is no longer a Prisoner's Dilemma. To do this, the payoff functions of

each player should be changed in order to make it unprofitable for anyone to defect. In

the case of the example given above, the payoff functions of each individual would

change if there were a fine for watering the garden during a drought. Such a solution is

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Game Theory

considered a "political" solution and oftentimes these sorts of solutions cannot be

implemented.

Parfit argues that an even better solution would be to find ways to make people cooperate

for purely moral reasons. Parfit proposes that the way to achieve such a "moral" solution

would be to educate society about the Prisoner's Dilemma and it's most desirable, though

irrational solution.

Kant's Categorical Imperative

Immanuel Kant's categorical imperative, which is intended to be a fundamental principle

of morality, states: "Act only on such a maxim through which you can at the same time

will that it should become a universal law." A maxim is just a personal rule of conduct

while the universal law is the conduct of all people. Kant's categorical imperative is

continually debated among moral philosophers because of its obscurity. Through the use

of Game Theory, Kant's views can be clarified. Kant's beliefs, when understood, offers a

moral solution to the Prisoner's Dilemma.

One of Kant's examples of categorical imperative is illustrated in the following maxim:

"Always borrow money when in need and promise to pay it back without any intention

of keeping the promise." This maxim cannot possibly made into a universal law because

it cannot be made universal without creating a contradiction. That is, if this maxim was

made universal, then everyone would break promises and a promise would have no

meaning and therefore promises would cease to exist. Therefore, if this maxim were

made universal, a logical contradiction would follow.

In terms of Game Theory, Kant's categorical imperative can be restated as follows:

"Choose only a strategy which, if you could will it to be chosen by all the players, would

yield a better outcome from you point of view than any other". This statement, then,

becomes a solution to the Prisoner's Dilemma. That is, according to Kant's categorical

imperative, only a cooperative choice can result. This is because the personal choice of

defecting, if made universal, is in contradiction to one's personal interest (similar to the

above example).

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Game Theory

Hobbes's and Rousseau's Social Contract

Through the use of Game Theory, Hobbes' argument, later made popular by Jean-

Jacques Rousseau, for absolute monarchy can be reconstructed. Hobbes argued that,

without some form of external constraint on people's behaviors, anarchy would ensue.

Cooperation among people would be impossible since people act only to maximize

individual welfare and not the welfare of society as a whole. Granted, there will exists

altruists (maybe even many of them) who constrain their self-interests for the good of

others. However, if even one self-interested person exists, he/she will exploit the

altruists' constraints, profiting from both his/her absence of constraint and the altruist's

unselfish behavior. As a result, Hobbes believes that it is psychologically unnatural for

altruists to exist. If just one narrowly self-interested person exists no altruist can survive

unless he/she becomes narrowly self-interested too. In such an environment, known as a

State of Nature, Hobbes argues that a person must always be suspicious that another will

attack in order to maximize his/her own self-interest. Therefore, in order for a person to

maximize his best interest, he must attack the other person before that other person can

attack. Each such conflict between two people in a state of nature has been termed as the

"Hobbesian Dilemma." However, in the field of Game Theory, the Hobbesian Dilemma

has the same structure as a "Prisoner's Dilemma."

Hobbes believed that the "Hobbesian Dilemma" results in a State of Nature because

morality is an unstable enforcer of social cooperation. According to Hobbes, a stable

enforcer can only exist if not one person can deviate from the established rule by which

the rest adhere to. Since cooperation among people is biologically necessary, a stable

enforcer must exist. Hobbes believes that the best form of social enforcement is the

existence of an all-powerful sovereign.

Resource Allocation and Networking

Computer network bandwidth can be viewed as a limited resource. The users on the

network compete for that resource. Their competition can be simulated using game

theory models. No centralized regulation of network usage is possible because of the

diverse ownership of network resources.

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Game Theory

The problem is of ensuring the fair sharing of network resources. For example, ten

Stanford students on the same local network need access to the Internet. Each person, by

using their network connection, diminishes the quality of the connection for the other

users. This particular case is that of a volunteer's dilemma. That is, if one person abstains

from using the network, the other people will be better off, but that person will be worse

off.

If a centralized system could be developed which would govern the use of the shared

resources, each person would get an assigned network usage time or bandwidth, thereby

limiting each person's usage of network resources to his or her fair share.

As of yet, however, such a system remains an impossibility, making the situation of

sharing network resources a competitive game between the users of the network and

decreasing everyone's utility.

Biology

Although the natural world is often thought of as brutal, dog-eat-dog type, cooperation

exists between many different species. The reason behind this coexistence can be

modeled using game theory. For example, birds called ziczacs enter crocodiles' mouths

to eat parasites. This symbiosis allows crocodiles to achieve good oral hygiene and

allows the ziczacs to get a decent meal. But any crocodile can easily eat a ziczac (defect).

So why don't they? Apparently, over the eons of evolutionary action, the crocodiles and

the ziczacs have learned the benefits of cooperation, the "equilibrium point."

Of course, chances are that neither the crocodiles nor the ziczacs rationalize their

behavior with game theory. But their behavior can still be modeled using game theory

principles.

Artificial Intelligence

One of the marks that differentiate a human from a machine is the human's ability to

make independent decisions based on environmental stimuli. Most computer programs

that are required to make any sort of a decision are currently pre-programmed with the

lists of decisions based on a number of conditions. However, if those conditions are not

36

Game Theory

met in some way or are altered, computers have no way of making decisions they were

not programmed to make.

In the future, AI programs may be endowed with the ability to make new decisions

unplanned for by their creators. This would require the programs to be able to generate

new payoff matrices based on the observed stimuli and experience. A program that is

able to do that would be capable of learning and would, in a lot of ways, resemble the

human decision-making process.

Economics

Many of the interactions in the business world may be modeled using game theory

methodology. A famous example is that of the similarity of the price-setting of

oligopolies to the Prisoner's Dilemma. If an oligopoly situation exists, the companies are

able to set prices if they choose to cooperate with each other. If they cooperate, both are

able to set higher prices, leading to higher profits. However, if one company decides to

defect by lowering its price, it will get higher sales, and, consequently, bigger profits

than its competitor(s), who will receive lower profits. If both companies decide to defect,

i.e. lower prices, a price war will ensue, in which case neither company will profit, since

it will retain its market share and experience lower revenues at the same time.

Similar arguments can be extended to many cases like:

advertising for a company’s products

international trade between nations and Trade Barriers

expenditure on National Defense for two neighbouring nations

This can better be explained with the help of the following example:

Suppose PepsiCo & Coca-Cola enter a new market and decide to form a cartel and fix

prices. Assuming that if they honour the cartel agreement, they both would land up with

revenues of $6 million each. But in case one of them cheats and the other does not, the

difference in revenues could be huge ($6 million in this case; between the companies).

37

Game Theory

The temptation to cheat is very high in this case and if both the companies cheat on the

agreement, then it’s a loss to both as can be very well seen from the following table.

PepsiCo

Cheat on Cartel

(Charge Low Price)

Don’t Cheat

(Charge Monopoly Price)

Coca-

Cola

Cheat on

Cartel$3 million each

Coke earns $8 million

Pepsi earns $2 million

Don’t

Cheat

Coke earns $2 million

Pepsi earns $8 million$6 million each

Prisoner's dilemma is not the only game theory model which can be used to model

economic situations. Other models can be applied to different situations and, in many

cases, can suggest the best outcome for all parties concerned.

38

Game Theory

Case Study: Limitations of ‘Game

Theory’

& the ‘Theory of Moves’

Theory of Moves

"We're eyeball to eyeball, and I think the other fellow just

blinked" were the eerie words of Secretary of State Dean Rusk at

the height of the Cuban missile crisis in October 1962. He was

referring to signals by the Soviet Union that it desired to defuse

the most dangerous nuclear confrontation ever to occur between

the superpowers, which many analysts have interpreted as a

classic instance of nuclear "Chicken".

Chicken is the usual game used to model conflicts in which the players are on a collision

course. The players may be drivers approaching each other on a narrow road, in which

each has the choice of swerving to avoid a collision or not swerving. In the novel Rebel

without a Cause, which was later made into a movie starring James Dean, the drivers

were two teenagers, but instead of bearing down on each other they both raced toward a

cliff, with the object being not to be the first driver to slam on his brakes and thereby

"chicken out", while, at the same time, not plunging over the cliff.

While ostensibly a game of Chicken, the Cuban missile crisis is in fact not well modelled

by this game. Another game more accurately represents the preferences of American and

Soviet leaders, but even for this game standard game theory does not explain their

choices.

On the other hand, the "theory of moves," which is founded on game theory but radically

changes its standard rules of play, does retrodict, or make past predictions of, the leaders'

choices. More important, the theory explicates the dynamics of play, based on the

assumption that players think not just about the immediate consequences of their actions

but their repercussions for future play as well.

39

Game Theory

I will use the Cuban missile crisis to illustrate parts of the theory, which is not just an

abstract mathematical model but one that mirrors the real-life choices, and underlying

thinking, of flesh-and-blood decision makers. Indeed, Theodore Sorensen, special

counsel to President John Kennedy, used the language of "moves" to describe the

deliberations of Excom, the Executive Committee of key advisors to Kennedy during the

Cuban missile crisis:

"We discussed what the Soviet reaction would be to any possible move by the United

States, what our reaction with them would have to be to that Soviet action, and so on,

trying to follow each of those roads to their ultimate conclusion."

Classical Game Theory and the Missile Crisis

Game theory is a branch of mathematics concerned with decision-making in social

interactions. It applies to situations (games) where there are two or more people (called

players) each attempting to choose between two more more ways of acting (called

strategies). The possible outcomes of a game depend on the choices made by all players,

and can be ranked in order of preference by each player.

In some two-person, two-strategy games, there are combinations

of strategies for the players that are in a certain sense "stable".

This will be true when neither player, by departing from its

strategy, can do better. Two such strategies are together known

as ‘Nash equilibrium’, named after John Nash, a mathematician

who received the Nobel prize in economics in 1994 for his work

on game theory.

Nash equilibria do not necessarily lead to the best outcomes for one, or even both,

players. Moreover, for the games that will be analyzed - in which players can only rank

outcomes ("ordinal games") but not attach numerical values to them ("cardinal games") -

they may not always exist. (While they always exist, as Nash showed, in cardinal games,

Nash equilibria in such games may involve "mixed strategies," which will be described

later.)

The Cuban missile crisis was precipitated by a Soviet attempt in October 1962 to install

medium-range and intermediate-range nuclear-armed ballistic missiles in Cuba that were

40

Game Theory

capable of hitting a large portion of the United States. The goal of the United States was

immediate removal of the Soviet missiles, and U.S. policy makers seriously considered

two strategies to achieve this end [see Figure 1 below]:

1. A naval blockade (B), or "quarantine" as it was euphemistically called, to

prevent shipment of more missiles, possibly followed by stronger action to

induce the Soviet Union to withdraw the missiles already installed.

2. A "surgical" air strike (A) to wipe out the missiles already installed,

insofar as possible, perhaps followed by an invasion of the island.

The alternatives open to Soviet policy makers were:

1. Withdrawal (W) of their missiles.

2. Maintenance (M) of their missiles.

Soviet Union (S.U.)

Withdrawal (W) Maintenance (M)

United

States

(U.S.)

Blockade

(B)

Compromise

(3,3)

Soviet victory,

U.S. defeat

(2,4)

Air strike

(A)

U.S. victory,

Soviet defeat

(4,2)

Nuclear war

(1,1)

Figure 1: Cuban missile crisis as Chicken

Key: (x, y) (payoff to U.S., payoff to S.U.)

4best; 3next best; 2next worst; 1worst

Nash equilibria underscored

These strategies can be thought of as alternative courses of action that the two sides, or

"players" in the parlance of game theory, can choose. They lead to four possible

outcomes, which the players are assumed to rank as follows: 4=best; 3=next best; 2=next

worst; and l=worst. Thus, the higher the number, the greater the payoff; but the payoffs

are only ordinal, that is, they indicate an ordering of outcomes from best to worst, not the

degree to which a player prefers one outcome over another. The first number in the

41

Game Theory

ordered pairs for each outcome is the payoff to the row player (United States), the second

number the payoff to the column player (Soviet Union).

Needless to say, the strategy choices, probable outcomes, and associated payoffs shown

in Figure 1 provide only a skeletal picture of the crisis as it developed over a period of

thirteen days. Both sides considered more than the two alternatives listed, as well as

several variations on each. The Soviets, for example, demanded withdrawal of American

missiles from Turkey as a quid pro quo for withdrawal of their own missiles from Cuba,

a demand publicly ignored by the United States.

Nevertheless, most observers of this crisis believe that the two superpowers were on a

collision course, which is actually the title of one book describing this nuclear

confrontation. They also agree that neither side was eager to take any irreversible step,

such as one of the drivers in Chicken might do by defiantly ripping off the steering wheel

in full view of the other driver, thereby foreclosing the option of swerving.

Although in one sense the United States "won" by getting the Soviets to withdraw their

missiles, Premier Nikita Khrushchev of the Soviet Union at the same time extracted from

President Kennedy a promise not to invade Cuba, which seems to indicate that the

eventual outcome was a compromise of sorts. But this is not game theory's prediction for

Chicken, because the strategies associated with compromise do not constitute a Nash

equilibrium.

To see this, assume play is at the compromise position (3,3), that is, the U.S. blockades

Cuba and the S.U. withdraws its missiles. This strategy is not stable, because both

players would have an incentive to defect to their more belligerent strategy. If the U.S.

were to defect by changing its strategy to airstrike, play would move to (4,2), improving

the payoff the U.S. received; if the S.U. were to defect by changing its strategy to

maintenance, play would move to (2,4), giving the S.U. a payoff of 4. (This classic game

theory setup gives us no information about which outcome would be chosen, because the

table of payoffs is symmetric for the two players. This is a frequent problem in

interpreting the results of a game theoretic analysis, where more than one equilibrium

position can arise.) Finally, should the players be at the mutually worst outcome of (1,1),

that is, nuclear war, both would obviously desire to move away from it, making the

strategies associated with it, like those with (3,3), unstable. 42

Game Theory

Theory of Moves and the Missile Crisis

Using Chicken to model a situation such as the Cuban missile

crisis is problematic not only because the (3,3) compromise

outcome is unstable but also because, in real life, the two sides did

not choose their strategies simultaneously, or independently of

each other, as assumed in the game of Chicken described above.

The Soviets responded specifically to the blockade after it was

imposed by the United States. Moreover, the fact that the United

States held out the possibility of escalating the conflict to at least an air strike indicates

that the initial blockade decision was not considered final - that is, the United States

considered its strategy choices still open after imposing the blockade.

As a consequence, this game is better modelled as one of sequential bargaining, in which

neither side made an all-or-nothing choice but rather both considered alternatives,

especially should the other side fail to respond in a manner deemed appropriate. In the

most serious breakdown in the nuclear deterrence relationship between the superpowers

that had persisted from World War II until that point, each side was gingerly feeling its

way, step by ominous step. Before the crisis, the Soviets, fearing an invasion of Cuba by

the United States and also the need to bolster their international strategic position,

concluded that installing the missiles was worth the risk. They thought that the United

States, confronted by a fait accompli, would be deterred from invading Cuba and would

not attempt any other severe reprisals. Even if the installation of the missiles precipitated

a crisis, the Soviets did not reckon the probability of war to be high (President Kennedy

estimated the chances of war to be between 1/3 and 1/2 during the crisis), thereby

making it rational for them to risk provoking the United States.

There are good reasons to believe that U.S. policymakers did not view the confrontation

to be Chicken-like, at least as far as they interpreted and ranked the possible outcomes. I

offer an alternative representation of the Cuban missile crisis in the form of a game I will

call Alternative, retaining the same strategies for both players as given in Chicken but

presuming a different ranking and interpretation of outcomes by the United States [see

Figure 2]. These rankings and interpretations fit the historical record better than those of

"Chicken", as far as can be told by examining the statements made at the time by

43

Game Theory

President Kennedy and the U.S. Air Force, and the type and number of nuclear weapons

maintained by the S.U. (more on this below).

BW: The choice of blockade by the United States and withdrawal by the Soviet Union

remains the compromise for both players - (3,3).

BM: In the face of a U.S. blockade, Soviet maintenance of their missiles leads to a

Soviet victory (its best outcome) and U.S. capitulation (its worst outcome) - (1,4).

AM: An air strike that destroys the missiles maintained by Soviets is an "honourable"

U.S. action (its best outcome) and thwarts the Soviets (their worst outcome) - (4,1).

AW: An air strike that destroys the missiles that the Soviets were withdrawing is a

"dishonorable" U.S. action (its next-worst outcome) and thwarts the Soviets (their next-

worst outcome) - (2,2).

Soviet Union (S.U.)

Withdrawal (W) Maintenance (M)

United

States

(US)

Blockade

(B)

Compromise

(3,3)

Soviet victory,

U.S. capitulation

(1,4)

Air strike

(A)

"Dishonourable" U.S.

action,

Soviets thwarted

(2,2)

"Honourable" U.S.

action,

Soviets thwarted

(4,1)

Figure 2: Cuban missile crisis as Alternative

Key: (x, y) (payoff to U.S., payoff to S.U.)

4best; 3next best; 2next worst; 1worst

Nonmyopic equilibria in bold

Arrows indicate direction of cycling

Even though an air strike thwarts the Soviets at both outcomes (2,2) and (4,1), I interpret

(2,2) to be less damaging for the Soviet Union. This is because world opinion, it may be

surmised, would severely condemn the air strike as a flagrant overreaction - and hence a

"dishonourable" action of the United States - if there were clear evidence that the Soviets

44

Game Theory

were in the process of withdrawing their missiles anyway. On the other hand, given no

such evidence, a U.S. air strike, perhaps followed by an invasion, would action to

dislodge the Soviet missiles.

The statements of U.S. policy makers support Alternative. In responding to a letter from

Khrushchev, Kennedy said,

"If you would agree to remove these weapons systems from Cuba . . . we, on our part,

would agree . . . (a) to remove promptly the quarantine measures now in effect and (b) to

give assurances against an invasion of Cuba,"

which is consistent with Alternative since (3,3) is preferred to (2,2) by the United States,

whereas (4,2) is not preferred to (3,3) in Chicken.

If the Soviets maintained their missiles, the United States preferred an air strike to the

blockade. As Robert Kennedy, a close adviser to his brother during the crisis, said,

"If they did not remove those bases, we would remove them,"

which is consistent with Alternative, since the United States prefers (4,1) to (1,4) but not

(1,1) to (2,4) in Chicken.

Finally, it is well known that several of President Kennedy's advisers felt very reluctant

about initiating an attack against Cuba without exhausting less belligerent courses of

action that might bring about the removal of the missiles with less risk and greater

sensitivity to American ideals and values. Pointedly, Robert Kennedy claimed that an

immediate attack would be looked upon as

"a Pearl Harbor in reverse, and it would blacken the name of the United States in the

pages of history," which is again consistent with the Alternative since the United States

ranks AW next worst (2) - a "dishonourable" U.S. action - rather than best (4) - a U.S.

victory - in Chicken.

If Alternative provides a more realistic representation of the participants' perceptions

than Chicken does, standard game theory offers little help in explaining how the (3,3) 45

Game Theory

compromise was achieved and rendered stable. As in Chicken, the strategies associated

with this outcome are not a Nash equilibrium, because the Soviets have an immediate

incentive to move from (3,3) to (1,4).

However, unlike Chicken, Alternative has no outcome at all that is a Nash equilibrium,

except in "mixed strategies". These are strategies in which players randomize their

choices, choosing each of their two so-called pure strategies with specified probabilities.

But mixed strategies cannot be used to analyse Alternative, because to carry out such an

analysis, there would need to be numerical payoffs assigned to each of the outcomes, not

the rankings I have assumed.

The instability of outcomes in Alternative can

most easily be seen by examining the cycle of

preferences, indicated by the arrows going in

a clockwise direction in this game. Following

these arrows shows that this game is cyclic,

with one player always having an immediate

incentive to depart from every state: the

Soviets from (3,3) to (1,4); the United States

from (1,4) to (4,1); the Soviets from (4,1) to

(2,2); and the United States from (2,2) to

(3,3). Again we have indeterminacy, but not because of multiple Nash equilibria, as in

Chicken, but rather because there are no equilibria in pure strategies in Alternative.

Rules of Play in Theory of Moves

How, then, can we explain the choice of (3,3) in Alternative, or Chicken for that matter,

given its nonequilibrium status according to standard game theory? It turns out that (3,3)

is a "nonmyopic equilibrium" in both games, and uniquely so in Alternative, according to

the theory of moves (TOM). By postulating that players think ahead not just to the

immediate consequences of making moves, but also to the consequences of

countermoves to these moves, counter-countermoves, and so on, TOM extends the

strategic analysis of conflict into the more distant future.

46

Game Theory

To be sure, game theory allows for this kind of thinking through the analysis of "game

trees," where the sequential choices of players over time are described. But the game tree

continually changed with each development in the crisis. By contrast, what remained

more or less constant was the configuration of payoffs of Alternative, though where the

players were in the matrix changed. In effect, TOM, by describing the payoffs in a single

game but allowing players to make successive calculations of moves to different

positions within it, adds nonmyopic thinking to the economy of description offered by

classical game theory.

The founders of game theory, John von Neumann and Oskar Morgenstern, defined a

game to be "the totality of rules of play which describe it." While the rules of TOM apply

to all games between two players, here I will assume that the players each have just two

strategies. The four rules of play of TOM describe the possible choices of the players at

each stage of play:

Rules of Play

1. Play starts at an initial state, given at the intersection of the row and column of a

payoff matrix.

2. Either player can unilaterally switch its strategy, or make a move, and thereby

change the initial state into a new state, in the same row or column as the initial

state. The player who switches is called player l (P1).

3. Player 2 (P2) can respond by unilaterally switching its strategy, thereby moving

the game to a new state.

4. The alternating responses continue until the player (P1 or P2) whose turn it is to

move next chooses not to switch its strategy. When this happens, the game

terminates in a final state, which is the outcome of the game.

Termination Rule

5. A player will not move from an initial state if this moves (i) leads to a less

preferred outcome, or (ii) returns play to the initial state, making this state the

outcome.

47

Game Theory

Precedence Rule

6. If it is rational for one player to move and the other player not to move from the

initial state, the move takes precedence: it overrides staying, so the outcome will

be induced by the player that moves.

Note that the sequence of moves and countermoves is strictly alternating: first, say, the

row player moves, then the column player, and so on, until one player stops, at which

point the state reached is final and, therefore, the outcome of the game. I assume that no

payoffs accrue to players from being in a state unless it becomes the outcome (which

could be the initial state if the players choose not to move from it).

To assume otherwise would require that payoffs be numerical, rather than ordinal ranks,

which players can accumulate as they pass through states. But in many real-life games,

payoffs cannot easily be quantified and summed across the states visited. Moreover, the

big reward in many games depends overwhelmingly on the final state reached, not on

how it was reached. In politics, for example, the payoff for most politicians is not in

campaigning, which is arduous and costly, but in winning.

Rule 1 differs drastically from the corresponding rule of play in standard game theory, in

which players simultaneously choose strategies in a matrix game that determines its

outcome. Instead of starting with strategy choices, TOM assumes that players are already

in some state at the start of play and receive payoffs from this state only if they stay.

Based on these payoffs, they must decide, individually, whether or not to change this

state in order to try to do better.

Of course, some decisions are made collectively by players, in which case it is

reasonable to say that they choose strategies from scratch, either simultaneously or by

coordinating their choices. But if, say, two countries are coordinating their choices, as

when they agree to sign a treaty, the important strategic question is what individualistic

calculations led them to this point. The formality of jointly signing the treaty is the

culmination of their negotiations and does not reveal the move-countermove process that

preceded the signing. It is precisely these negotiations, and the calculations underlying

them, that TOM is designed to uncover.

48

Game Theory

To continue this example, the parties that sign the treaty were in some prior state from

which both desired to move - or, perhaps, only one desired to move and the other could

not prevent this move from happening (rule 6). Eventually they may arrive at a new state,

after, say, treaty negotiations, in which it is rational for both countries to sign the treaty

that was previously negotiated.

As with a treaty signing, almost all outcomes of games that we observe have a history.

TOM seeks to explain strategically the progression of (temporary) states that lead to a

(more permanent) outcome. Consequently, play of a game starts in an initial state, at

which players collect payoffs only if they remain in that state so that it becomes the final

state, or outcome, of the game.

If they do not remain, they still know what payoffs they would have collected had they

stayed; hence, they can make a rational calculation of the advantages of staying or

moving. They move precisely because they calculate that they can do better by switching

strategies, anticipating a better outcome when the move-countermove process finally

comes to rest. The game is different, but not the configuration of payoffs, when play

starts in a different state.

Rules 1 - 4 (rules of play) say nothing about what causes a game to end, only when:

termination occurs when a "player whose turn it is to move next chooses not to switch its

strategy" (rule 4). But when is it rational not to continue moving, or not to move at all

from the initial state?

Rule 5 (termination rule) says when a player will not move from an initial state. While

condition (i) of this rule needs no defence, condition (ii) requires justification. It says that

if it is rational, after P1 moves, for play of the game to cycle back to the initial state, P1

will not move in the first place. After all, what is the point of initiating the move-

countermove process if play simply returns to "square one," given that the players

receive no payoffs along the way to the outcome?

Backward Induction

49

Game Theory

To determine where play will end up when at least one player wants to move from the

initial state, I assume the players use backward induction. This is a reasoning process by

which the players, working backward from the last possible move in a game, anticipate

each other's rational choices. For this purpose, I assume that each has complete

information about the other's preferences, so each can calculate the other player's rational

choices, as well as its own, in deciding whether to move from the initial state or any

subsequent state.

To illustrate backward induction, consider again the game Alternative in Figure 2. After

the missiles were detected and the United States imposed a blockade on Cuba, the game

was in state BM, which is worst for the United States (1) and best for the Soviet Union

(4). Now consider the clockwise progression of moves that the United States can initiate

by moving to AM, the Soviet Union to AW, and so on, assuming the players look ahead

to the possibility that the game makes one complete cycle and returns to the initial state

(state 1):

State 1 State 2 State 3 State 4 State 1

U.S. startsU.S.

(1,4)

S.U.

(4,1)

U.S.

(2,2)|

S.U.

(3,3)

(1,4)

Survivor (2,2) (2,2) (2,2) (1,4)

This is a game tree, though drawn horizontally rather than vertically. The survivor is a

state selected at each stage as the result of backward induction. It is determined by

working backward from where play, theoretically, can end up (state 1, at the completion

of the cycle).

Assume the players' alternating moves have taken them clockwise in Alternative from

(1,4) to (4,1) to (2,2) to (3, 3), at which point S.U. in state 4 must decide whether to stop

at (3,3) or complete the cycle by returning to (1,4). Clearly, S.U. prefers (1,4) to (3,3), so

(1,4) is listed as the survivor below (3,3): because S.U. would move the process back to

(1,4) should it reach (3,3), the players know that if the move-countermove process

reaches this state, the outcome will be (1,4).

50

Game Theory

Knowing this, would U.S. at the prior state, (2,2), move to (3,3)? Because U.S. prefers

(2,2) to the survivor at (3,3) - namely, (1,4) - the answer is no. Hence, (2,2) becomes the

survivor when U.S. must choose between stopping at (2,2) and moving to (3,3) - which,

as I just showed, would become (1,4) once (3,3) is reached.

At the prior state, (4,1), S.U. would prefer moving to (2,2) than stopping at (4,1), so (2,2)

again is the survivor if the process reaches (4,1). Similarly, at the initial state, (1,4),

because U.S. prefers the previous survivor, (2,2), to (1,4), (2,2) is the survivor at this

state as well.

The fact that (2,2) is the survivor at the initial state, (1,4), means that it is rational for

U.S. to move to (4,1), and S.U. subsequently to (2,2), where the process will stop,

making (2,2) the rational choice if U.S. moves first from the initial state, (1,4). That is,

after working backwards from S.U.'s choice of completing the cycle or not from (3,3),

the players can reverse the process and, looking forward, determine what is rational for

each to do. I indicate that it is rational for the process to stop at (2,2) by the vertical line

blocking the arrow emanating from (2,2), and underscoring (2,2) at this point.

Observe that (2,2) at state AM is worse for both players than (3,3) at state BW. Can S.U.,

instead of letting U.S. initiate the move-countermove process at (1,4), do better by

seizing the initiative and moving, counterclockwise, from its best state of (1,4)? Not only

is the answer yes, but it is also in the interest of U.S. to allow S.U. to start the process, as

seen in the following counterclockwise progression of moves from (1,4):

State 1 State 2 State 3 State 4 State 1

S.U. startsS.U.

(1,4)

U.S.

(3,3) |

S.U.

(2,2)

U.S.

(4,1)

(1,4)

Survivor (3,3) (3,3) (2,2) (4,1)

51

Game Theory

S.U., by acting "magnanimously" in moving from victory (4) at BM to compromise (3)

at BW, makes it rational for U.S. to terminate play at (3,3), as seen by the blocked arrow

emanating from state 2. This, of course, is exactly what happened in the crisis, with the

threat of further escalation by the United States, including the forced surfacing of Soviet

submarines as well as an air strike (the U.S. Air Force estimated it had a 90 percent

chance of eliminating all the missiles), being the incentive for the Soviets to withdraw

their missiles.

Applying TOM

Like any scientific theory, TOM's calculations may not take

into account the empirical realities of a situation. In the

second backward-induction calculation, for example, it is

hard to imagine a move by the Soviet Union from state 3 to

state 4, involving maintenance (via reinstallation?) of their

missiles after their withdrawal and an air strike. However, if

a move to state 4, and later back to state 1, were ruled out as

infeasible, the result would be the same: commencing the

backward induction at state 3, it would be rational for the Soviet Union to move initially

to state 2 (compromise), where play would stop.

Compromise would also be rational in the first backward-induction calculation if the

same move (a return to maintenance), which in this progression is from state 4 back to

state 1, were believed infeasible: commencing the backward induction at state 4, it would

be rational for the United States to escalate to air strike to induce moves that carry the

players to compromise at state 4. Because it is less costly for both sides if the Soviet

Union is the initiator of compromise - eliminating the need for an air strike - it is not

surprising that this is what happened.

To sum up, the Theory of Moves renders game theory a more dynamic theory. By

postulating that players think ahead not just to the immediate consequences of making

moves, but also to the consequences of countermoves to those moves, counter-

countermoves, and so on, it extends the strategic analysis of conflicts into the more

distant future. TOM has also been used to elucidate the role that different kinds of power

- moving, order and threat - may have on conflict outcomes, and to show how

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Game Theory

misinformation can affect player choices. These concepts and the analysis have been

illustrated by numerous cases, ranging from conflicts in the Bible to disputes and

struggles today.

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Game Theory

Conclusion

"Managers have much to learn from game theory - provided they use it to clarify their

thinking, not as a substitute for business experience" 

FOR old-fashioned managers, business was a branch of warfare - a way of 'capturing

markets' and 'making a killing'. Today, however, the language is all about working with

suppliers, building alliances, and thriving on trust and loyalty. Management theorists like

to point out that there is such a thing as 'win-win', and that business feuds can end up

hurting both parties. 

But this can be taken too far. Microsoft's success has helped Intel, but it has been hell for

Apple Computer. Instead, business needs a new way of thinking that makes room for

collaboration as well as competition, for mutual benefits as well as trade-offs. Enter

game theory. 

Stripped to its essentials, game theory is a tool for understanding how decisions affect

each other. Until the theory came along, economists assumed that firms could ignore the

effects of their behaviour on the actions of rivals, which was fine when competition was

perfect or a monopolist held sway, but was otherwise misleading. Game theorists argue

that firms can learn from game players: no card player plans his strategy without

thinking about how other players are planning theirs. 

Economists have long used game theory to illuminate practical problems, such as what to

do about global warming or about fetuses with Down's syndrome. Now business people

have started to wake up to the theory's possibilities. McKinsey, a consultancy, is setting

up a practice in game theory. Firms as diverse as Xerox, an office-equipment maker,

Bear Stearns, an investment bank, and PepsiCo, a soft-drinks giant, are all interested.

They will no doubt seize on 'Co-opetition' (Doubleday, $ 24.95), because it is written by

two of the leading names in the field, Adam Brandenburger, of Harvard Business School,

and Barry Nalebuff, of the Yale School of Management. It also helps by using readable

case studies rather than complex mathematics. 

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Game Theory

The main practical use of game theory, say the authors, is to help a firm decide when to

compete and when to co-operate. Broadly speaking, the time to co-operate is when you

are increasing the size of the pie, and the time to compete is when you are dividing it up.

The authors also argue that, to get a full picture of their business, managers need to think

about a new category of firms, 'complementers', which lead your customers to value your

products more highly than if they had only your product. Hot-dog makers and Colman's

mustard are complementers: buy one and you are more likely to buy the other. So are

Intel and Microsoft. 

The most important thing to know about a game is who the players are. A small change

in the number of players can have unexpected consequences. The Holland Sweetener

Company, a Dutch-Japanese joint venture, discovered this to its cost in the late 1980s

when it tried to break NutraSweet's monopoly of the American artificial-sweetener

market. NutraSweet managed to keep the predator out, but only after Coca-Cola and

Pepsi used the threat of competition to force NutraSweet to lower its prices. 

When competition between two players benefits third parties in this way, there is scope

for the beneficiary to split its gains. Holland Sweetener in effect gave up its share of the

gains that it had helped Coke and Pepsi to win. BellSouth, a telephone company, was

wiser: it insisted on being paid to play. The firm said that it would bid against Craig

McCaw for control of LIN Broadcasting Corporation only if LIN paid it $ 54m for

entering the fray and a further $ 15m in expenses if it lost the bid. 

One way for a player to do well in a game is to make itself indispensable. Nintendo built

its video-games business in the late 1980s by restricting software developers to making

five games each, keeping retailers on short rations, and doing much of the development

in-house. Nobody else had any bargaining power. By contrast, IBM stored up trouble for

itself in personal computers by allowing Microsoft and Intel to establish a lock on the

two most valuable bits of the business. 

A second technique is to tempt lots of competing players into the game - for instance by

increasing the prize. That is what American Express did in 1994 when it organised a

coalition with other big companies to purchase health care. The potential contract was so

large that a host of health-care providers got into a bidding war. A third technique is to 55

Game Theory

make intelligent use of a resource which is worth more to your customer than to you. In

1993 TWA lifted itself off the bottom of the airline league by tearing out several rows of

seats that were usually empty because the carrier was so unpopular, giving passengers

more leg-room - and making the airline popular once more. 

When to stop playing 

Game theory seems a fine way to analyse decisions retrospectively. But is it much help

in the heat of battle? The track record of grand ideas imported from other disciplines,

notably chaos theory, is not impressive. However, the game theorists have already

notched up some significant practical successes. The Federal Communications

Commission used the theory to help design its $ 7 billion auction of radio spectrum for

mobile phones - and hundreds of mobile-phone companies also used the theory to

formulate their bids. 

But, as Peter Scott-Morgan, a consultant with Arthur D. Little, points out, game theorists

are worryingly silent about the links between a company's strategy and its internal

capabilities. Today's most successful managers craft their strategies on the basis of

knowledge of their own companies, and devote at least as much thought to the question

of how blueprints will be translated into practice. At their worst, game theorists represent

a throwback to the days of such whiz-kids as Robert McNamara, chairman of Ford in

1960 and later defence secretary, who thought that rigorous analytical skills were the key

to success. 

Yet nothing can ever substitute for deep first-hand knowledge and experience. So,

however sophisticated the games that managers play may be, they will still need to get

their hands dirty.

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