Game Theory in Economics

by

Jihane Bettahi

October 27th, 2014MATH 89S: Game Theory and Democracy

Professor Hubert Bray

INTRODUCTION

“What I think he thinks”, “What I think he thinks I think”, these are phrases that we form

when we are playing a game and we want to figure out the thinking of other players in

order to know which moves or actions to take to maximize our chances of winning.

Indeed, games that involve two or more players require that each player think of how the

other player(s) might respond to his move; in other words, each player has to consider the

possible decisions, choices or moves of the other player(s) together with his decisions for

a better outcome. Similarly, in the economic world, the different economic agents

consider how the other economic agents would respond to a specific action before

making a firm decision. For instance, producers, when setting the price of a product,

consider how the consumers would respond to that price and how the other companies

would compete or cooperate. Therefore, the economic world is a game where the

economic agents are the players. Consequently, it would not be surprising if we study

economic behaviors from a game theory perspective.

This paper, through the use of games and mathematical theories, explores some aspects

of how game theory applies to the economic field. In this vein, the paper starts by

defining game theory, its elements and principles. Then, using the Prisoners’ dilemma

game and Nash’s equilibrium, the paper analyzes how firms can use game theory to set

their prices and output. Finally, the paper uses the ultimatum and centipede games

together with the mathematical method of backward induction to analyze bargaining and

negotiations.

2

GAME THEORY

Game theory is a model that studies strategic game situations where players interact

within a defined set of rules and where the result depends on the actions of two or more

players. Game theory also predicts the decisions and choices to be made by the players

for an optimal outcome (McNulty).

Game theory started by studying two-players zero-sum games where the gains of one

player are the losses of the other. It then developed to study games with multiple players

where mutual positive gains are possible (Dixit and Barry, 2008).

The Elements of Game Theory

Game theory involves three main elements:

Players: these are the decision makers in the game. Players can choose any move among

a set of choices.

Strategy: it is the course of action taken by a player.

Payoff: this is the return or payout of a player at the end of the game. It is also referred to

as utility (Thomson Learning, 2005).

A game also consists of an information set and an equilibrium point. The information set

represents the information available at a specific point of the game; and the equilibrium is

the point of the game when players achieved an outcome.

3

Furthermore, game theory makes the assumptions that the players are rational- that is that

players know what in their best interests and that their goal is to get the maximum

payoffs they can get in the game (Investopedia).

Putting all those elements together, game theory is about finding the strategy, which is

the sequence of actions a player should take to get the best payoffs in a game.

The Principles of Game Theory

Game theory is said to involve the following principles:

1. Each player has a set of choices he can choose from

2. All the moves of a player result in a win or lose outcome

3. The outcomes or results of the game are well-defined

4. The players have complete information on the rules of the game and the payoffs

of other players. However, they don’t know the decisions that will be taken by

other players in advance, hence they make assumptions about the other players’

choices.

5. The players are rational, that is that they choose what in their best interest and

what brings them greater benefit or advantage.

Using these principles, a player, in a game, should know his payoff probability, should

think about and make a guess on other players’ payoff probability and think about what

other players think of his own payoff probability (Simley).

4

Linking Game Theory to Economics

Game theory studies the situations where the decisions made by one player affect all the

other players in the game. In fact, the games addressed by game theory have the essential

common feature of interdependence; that is that the payoff for each player depends on the

decisions made or the strategies played by the other players. Indeed, players interact with

one another, such interaction creating opportunities for competition or cooperation.

Additionally, these interactions can either be sequential, when each player knows the

previous actions of other players, or simultaneous, and this is when the players make a

decision at the same time and hence are unaware about the decisions taken by other

players. When making a decision about which action or move to take, a player looks

ahead and anticipates how the other player will respond to his move, then calculates what

the best move is. Therefore, a player actually has to put himself in the shoes of the other

players to think the way they would think in response to an action or a move he would

take.

In the economic world, there are interactions between different economic parties such as

producers and consumers, sellers and buyers, and companies between one another. Each

player in a market has to consider how other players will respond to his actions before

deciding on any strategy; for instance producers consider how the consumers would

respond to a higher price for a specific product or how other competing producers would

respond. Thus, economic agents play a game that can be studied by game theory to

predict people’s response and determine the strategies that will bring the best payoffs

(Gittins, 2012).

5

ECONOMIC APPLICATIONS OF GAME THEORY: PRICING, COMPETITION

The Prisoners’ Dilemma

The prisoners’ dilemma is a classic game where two suspects A and B are put in separate

rooms and are questioned about whether they participated in a crime. Each suspect is

faced with a choice of either confessing (testifying against the other suspect) or not

confessing (keeping silent). If both A and B confess, they both get 2 years of

imprisonment. If A confesses, and B doesn’t confess, then A doesn’t go to jail and B gets

6 years of imprisonment and vice versa. Finally, if A and B both choose not to confess,

they both get 1 years of imprisonment. Suspects cannot communicate with one another,

but each of them will consider the likely choice of the other suspect while making their

independent decision of whether to confess or not. We represent the prisoners’ dilemma

by the following payoff matrix:

Confess Not confess

Confess 2, 2 0, 6

Not confess 6, 0 1, 1

Now, we want to know what is the strategy (confessing or not confessing) that holds a

better payoff for each prisoner. If we analyze the choices and the payoffs for each player,

we find that player B gets a better payoff if he confesses regardless of whether A

confesses or not. Hence, B would better confess; confession is his dominant strategy. The

same applies to A. Therefore, the equilibrium strategy, called Nash equilibrium, is for

both players to confess. However, both A and B get better payoff if they both keep silent;

6

meaning that a cooperative behavior where they admit nothing will get them both a better

outcome (Riley, 2012).

Nash Equilibrium

Nash equilibrium refers to the set of strategies in a non-cooperative game such that none

of the players have an incentive to deviate from his strategy given the choice of other

players. In other words, none of the players can get an advantage from changing his

chosen strategy as long as the other players keep the same strategy. Simply put, players

are in Nash equilibrium if each of them made the best strategy choice considering the

strategy choices of the other players in the game. The concept of Nash equilibrium is

therefore very useful in predicting the simultaneous decisions to be made by people or

companies when the payoffs depend on the decisions of other players, people or

companies as Nash’s concept takes into account the decision-making of others in one’s

own decision making (Polak).

To express Nash’s equilibrium formally, we first consider the following notations:

SA = strategies available for player A (a Ì SA)

SB = strategies available for player B (b Ì SB)

UA = utility obtained by player A when particular strategies are chosen

UB = utility obtained by player B when particular strategies are chosen

When considering two-players games, two strategies (a*,b*) are in Nash equilibrium if

UA(a*,b*) ³ UA(a’,b*) for all a’ÌSA and UB(a*,b*) ³ Ub(a*,b’) for all b’ÌSB. That is that

a* is the best strategy for player A when B plays b* and vice versa (Thomson Learning,

2005).

7

More generally, we consider a game with n players where is the set of strategies for

player i and f is the utility or payoff function for each strategy. Considering that x(i) is the

strategy for player i and x(-i) is the strategy of all the other players, the strategy x* is a

Nash equilibrium if

Finally, we should note that not all the games have Nash equilibrium and that some

games have multiple Nash equilibriums. For instance, in the rock-paper-scissors game,

there isn’t Nash equilibrium because each strategy gives an incentive for the other player

to adopt another strategy.

Economic Applications: Pricing, Cournot Analysis, Bertrand Equilibrium

Considering the prisoners’ dilemma and Nash equilibrium, we now solve a pricing game

that depicts real economic situations. We consider two firms A and B that are to make a

decision between a high output or low output, the output being how much of a specific

product they should supply to the market. If both firm A and firm B choose a high output,

the supply in the market increases; hence the prices decrease and the profits are lower,

say $5m for each firm. But if they both choose low output, the supply will be limited, the

prices high and hence the profits will be higher, say $10m for each firm. If firm A

increases its output and firm B chooses a low output, then firm A gets $12m while firm B

gets $4m and vice versa (Riley, 2012). We represent these payoffs by the following

payoff matrix:

High output (B) Low output (B)

8

High output (A) 5m, 5m 12m, 4m

Low output (A) 4m, 12m 10m, 10m

Both firms want to maximize their profits. If we analyze the payoffs for each firm, we

find that firm A gets better payoffs if it chooses high output whether firm B opts for high

output or low output. In fact, if A chooses high output, it will get $5m profit if B also opts

for high output or it will get $12m profit if B opts for low output. However, if A chooses

low output, it can only get a better payoff if B also opts for low output; but every firm

will have an incentive to cheat on the other to get a higher profit. Therefore, A would

better choose high output. The same applies to B. Consequently, both firms choosing a

high output represent Nash equilibrium. However, the firms will get better payoffs if they

cooperate and decide on low output. Therefore a cooperative behavior is better for both

firms.

Looking at the same situation from another perspective, we introduce Cournot analysis.

In fact, the Cournot analysis is the situation where firms choose their output before the

price. The Cournot model predicts the price P’ of a product when the supply quantities

representing Nash equilibrium are defined. Indeed, once the level of output is determined,

the price P’ is that where the quantity demanded is equal to the quantity supplied by both

firms, and Nash equilibrium will only be achieved if PA = PB = P’, representing a

monopoly price (Thomson Learning, 2005).

We now consider the situation where the firms A and B are to decide on the price of a

specific product given a fixed output and a constant marginal cost c. The only condition

on the price is that it should exceed the marginal cost c. Since the two firms are

9

providing the same product, consumers will choose the product of the firm with the lower

price; hence the firm with the lower price “gains the entire market”. So every firm will

seek to have a price lower than that of the other firm. If firm A chooses a price PA greater

than the marginal cost c, then firm B chooses a price PB lower than PA; but if PB is greater

than c, then firm A will have an incentive to deviate from its initial price and lower the

price further than PB. Thus, Nash equilibrium will only be achieved when PA = PB = c, in

which case both firms equally share the market and none has an incentive to lower the

price further because c is already the minimum. Hence, the situation in hand results in a

competitive result. However, it is important to note that this competitive results holds

only under the assumptions that both firms have equal production cost and their products

are the same (perfect substitutes); should these assumptions change, the result will also

change. This pricing strategy is also called Bertrand equilibrium (Thomson learning,

2005).

Finally, when companies enter a business as a win-or-lose situation, their decisions are

evaluated through the two-players zero-sum game theory, which is a non-cooperative

model and hence implies perfect competition and duopoly in the market. For instance, the

competition between Coca-Cola and Pepsi, or Ford and General motors is a non-

cooperative model. However, in many other circumstances, companies want to get the

best payoffs rather than beat other companies, in which case cooperation and alliances

arise to achieve mutual benefit. In this context, the cooperation between Microsoft and

Intel is an eloquent example (Simley).

10

ECONOMIC APPLICATIONS OF GAME THEORY: BARGAINING

The ultimatum Game and The Centipede Game

We consider the game called the ultimatum game. In this game, two players A and B are

given a specific amount of money to divide between them. Player A starts and suggests

how the money should be divided between them. Player B responds by either accepting

or rejecting the proposal of player A. If player B accepts the proposal, the money is

divided as suggested by A. But if player B rejects the proposal, both players receive

nothing; the payoff being (0,0). For example, we suppose that players A and B are given

$10 dollars to split between them. Player A makes a proposal of how the $10 should be

divided. We assume that both players want to get the maximum possible amount of

money. Player A can offer any number between $0 (smallest) and $10 (largest) that

player B either accepts or rejects. The solution of the game is the best equilibrium

strategy for player A. But to solve the game, we start from the last player B. After A

makes an offer, since player B gets $0 from rejecting any offer, he will only have an

incentive to accept the offer if it gives him more than $0, that is any number between $1

and $10. Considering this information, player A decides what to offer. Player A can’t

keep all the 10$ because B will get $0 and will reject the offer. Since A, being rational,

wants to maximize the amount of money he gets, his next possible offer will give B as

little as possible but anything above $0, say A takes $9 and gives B $1. Since 1>0, B has

an incentive to accept the offer. If A deviates from this strategy, say by taking $8 and

giving $2 for instance, A will get less money. Hence, the equilibrium strategy for A to

11

maximize the money he gets is taking $9 and giving B $1. This is called the subgame

perfect equilibrium of the Ultimatum game. Practically, in real-word situations, this

strategy seems unfair to player B and hence is rejected even if rationality incentivizes

player B to accept the $1 (Spaniel).

Now, we consider another game called the centipede game. In the centipede game, a third

party puts two piles of money in front of two players, A and B. The players alternately

choose to either take the larger pile of money or pass the money. If a player chooses to

take the money, he gets the larger share of money, the other player gets the smaller one

and the game stops. If a player passes the money, the quantity of the piles is doubled and

the next player takes his turn to either take or pass the money and so on. If both players

always pass, they receive an equal amount of money at the end. We want to find the

equilibrium strategy for any player if the game has 100 rounds. To solve the game, we

start from the last round. Supposing that the two players are in the final round of the

game, the last player to play, say player B, will maximize the amount of money he gets if

he chooses to take the largest pile (otherwise, the money will be split equally between the

two players). If we suppose that the player B at the last round will take the largest pile,

player A will get more money if he takes the largest pile at the second-last round. Again,

with this information, player B will take the largest pile at the third-to-last round.

Following this backward reasoning, we conclude that any player will do better if he takes

the largest pile at the first round. Hence, the equilibrium strategy for any player is to take

the largest pile at any given opportunity, specifically, the first player should choose to

“take” at the first round. However, practically, given that the initial piles of money are

small, players tend to pass until the piles of money get bigger at later stages of the game

12

which make them get more money than they would have should they have taken the

money in the first round (Investopedia and EconPort, 2006).

Both games were solved through starting from the last move, or the end of the game to

determine earlier optimal moves in the game. This backward reasoning to find the

equilibrium strategy is called backward induction.

Backward Induction

Backward induction, as used in the previous games, consists of solving a game by

examining the best action or move of the last mover in the game; then assuming that the

last optimal action is taken, examining the next to last move in a similar way and so on

until getting to the beginning of the game; the first mover then chooses his action

depending on the best responses that follow. Therefore, backward induction is a

backward reasoning process that starts at the end of the problem or game to deduce the

optimal moves or actions to be taken in the game. In fact, backward induction follows

the algorithm below:

Take any final point in the game

Choose one of the payoff moves that give the mover at that final point of the game the highest payoff

Assign the best payoff to that point of the game

Eliminate all the other moves after that stage of the game

Check if there are non-terminal stages in the game

If no

If yes

13

Take the picked moves

Backward induction assumes that players are rational at each move and results in a

sequence of optimal moves that constitute the strategy profile. This strategy profile is

proven to be Nash equilibrium (MIT opencourseware).

Economic Application: Bargaining and Negotiations

We consider the economic situation of bargaining or negotiation. To determine the

outcome or result of bargaining, backward induction is to be used.

We suppose that two players own a dollar that they can only use after they split it. Hence,

the players negotiate or bargain about how they should split the dollar by making offers

and counteroffers to reach a consensus. We suppose that there are only two rounds of

negotiations (offer and counteroffer). Player A makes an offer of payoffs (x1,y1). Then,

player B either accepts or rejects the offer. If B accepts the offer, then the players get

payoffs (x1, y1). But if B rejects the offer, then B makes another offer that A either

accepts or rejects. If A rejects B’s proposal, then both players get nothing (0,0). We

analyze the problem using backward induction as we did in the ultimatum and centipede

games. In the second offer, if player A rejects the offer, he gets 0. Hence, player A will

accept any offer that gives him more than 0. Since player B wants to maximize his gain,

he will give the smallest amount possible to A that is greater than 0. So, if player B

rejects the first offer, he gets almost the whole dollar while player A gets almost nothing.

Thus, player B will only accept the first offer made by A if the offer gives him an amount

similar or greater than what B would get in the counter-offer round which is almost the

whole dollar (MIT opencourseware).

However, as in previous games, this might sound unfair and impractical. Thus, backward

14

induction seems to be failing. But it is important to note that these strategies hold under

the assumption that players only care about what in their best interest, that they will

accept anything greater than 0 and that the game is one or two-stages game. However, in

real life, people value fairness, can reject uneven low offers even if what they would get

from these offers is better than 0, and they alternate offers until one is accepted. In this

sense, we should carefully consider the assumptions or factors in general that affect any

bargaining situation. In fact, if we consider that there is a bargaining situation between

two players, say a buyer and a seller, where there are unlimited alternating offers; that is

to say that players keep alternating offers until an offer is accepted, where the value of

the dollar to be divided or the product to be sold doesn’t significantly change between

each round of the offers and where the players are equally impatient to buy or sell the

product, then we will find out that backward induction leads to an even and fair split or

agreement between the two players; but only under those assumptions and circumstances

will the outcome of bargaining be fair (Polak).

CONCLUSION

All in all, game theory, being a model that studies strategic situations where players

interact with one another and where the decisions of one player depend on that of the

others, is indeed an effective tool to analyze the strategies of the economic agents in the

economic world. Setting prices and output, competition and bargaining are only few

examples of the many economic applications of game theory. However, as we apply

game theory to the economic world, we should pay careful attention to the assumptions

that we make and how practical and real these assumptions are.

15

Summary

Game theory is a model that is used to analyze strategic situations where the decisions of one player depend on the decisions of the other players, interdependence being its main feature. Game theory involves three basic elements: players who are the decision-makers, strategies that are the decisions and actions of the players, and payoffs that are the results or outcomes of the game. Game theory also includes other elements that are the information set, the equilibrium point and a set of assumptions made such as the players being rational for instance. Game theory tightly relates to economics since the economic world also involves interactions between different economic agents (producers, consumers, firms…) where the decisions of an economic agent affect those of the others. Therefore, game theory has large applications in the economic field and can be used as a tool to study and predict economic behavior in a market.

Considering some economic applications of game theory, we analyze the classic game of the prisoners’ dilemma. In the game, we have two suspects that are being interrogated about a crime. The two suspects have two choices, either confess or not confess. Analyzing the game reveals that the equilibrium strategy is for both players to confess; this equilibrium strategy is called Nash equilibrium. Nash equilibrium is the set of strategies and actions taken by the players such that none of the players would be incentivized to change his strategy if he gets to know the strategy of the other players. That simply means that the players made their best strategy choice given the strategies of other players. The idea in the prisoners’ dilemma and the concept of Nash equilibrium have many applications in the economic field. When two firms want to decide on the output or price of a specific product, each firm considers how the other firm and the market would respond to the price or output chosen, which eventually lead the firms to Nash equilibrium and a decision where they either cooperate or compete.

Bargaining is another economic application of game theory. But before getting into its stakes, the paper analyzes the ultimatum and centipede games and introduces backward induction. In the ultimatum game, a player gives a take-it or leave-it offer to another player on how to divide 10 dollars (for example) between them. The responder to the offer can either accept or reject the proposed offer. If he rejects it, both players get nothing. Starting from the end of the game in a process of backward reasoning, the equilibrium strategy found is that the proposer should offer to take $9 and leave the other player with $1. A similar reasoning is applied to the centipede game to introduce the concept of backward induction. Backward induction is simply the process of starting from the end of the game to figure out a sequence of optimal moves for the game. This process is applied in bargaining. Applying backward induction to bargaining games might seem unfair and impractical, but it is important to note that this is due to the assumptions we make when analyzing bargaining situations from a game theory perspective such that the players being rational, accepting low offers and not valuing fairness. However, if we change our assumptions to match the reality, considering that there is an unlimited number of alternating offers, that the value of the product doesn’t significantly change over rounds and that the players are equally impatient, then backward induction leads to a fair and even agreement between the players.

16

Bibliography

"Backward Induction." EconPort. Experimental Economics Center, 2006. Web. 24 Oct. 2014. <http://www.econport.org/econport/request?page=man_gametheory_backinduct>.

"Backward Induction." Investopedia. Investopedia US, n.d. Web. 24 Oct. 2014. <http://www.investopedia.com/terms/b/backward-induction.asp>.

"Centipede Game." EconPort. Experimental Economics Center, 2006. Web. 24 Oct. 2014. <http://www.econport.org/econport/request?page=man_gametheory_exp_centipede>.

"Chapter 6 Nash Equilibrium." (n.d.): n. pag. MIT OpenCourseWare. MIT. Web. 22 Oct. 2014. <http://ocw.mit.edu/courses/economics/14-12-economic-applications-of-game-theory-fall-2012/lecture-notes/MIT14_12F12_chapter6.pdf>.

"Chapter 9 Backward Induction." (n.d.): n. pag. MIT OpenCourseWare. MIT. Web. 22 Oct. 2014. <http://ocw.mit.edu/courses/economics/14-12-economic-applications-of-game-theory-fall-2012/lecture-notes/MIT14_12F12_chapter9.pdf>.

"Chapter 10 Negotiations." (n.d.): n. pag. MIT OpenCourseWare. MIT. Web. 22 Oct. 2014. <http://ocw.mit.edu/courses/economics/14-12-economic-applications-of-game-theory-fall-2012/lecture-notes/MIT14_12F12_chapter10.pdf>.

"Chapter 15: Game Theory Models of Pricing." South-Western, a Division of Thomson Learning, 2005. Web. 24 Oct. 2014.

Dixit, Avinash, and Barry Nalebuff. "Game Theory." Library of Economics and Liberty. Liberty Fund, Inc, 2008. Web. 22 Oct. 2014. <http://www.econlib.org/library/Enc/GameTheory.html>.

Gittins, Ross. "How Economists Put Game Theory to Practical Use." The Sydney Morning Herald - Business Day. Fairfax Media, 20 Oct. 2012. Web. 22 Oct. 2014. <http://www.smh.com.au/business/how-economists-put-game-theory-to-practical-use-20121019-27wq1.html#ixzz3GpafXAZz>

McNulty, Daniel. "The Basics Of Game Theory." Investopedia. Investopedia US, n.d. Web. 22 Oct. 2014. <http://www.investopedia.com/articles/financial-theory/08/game-theory-basics.asp>.

"Nash Equilibrium." Wikipedia. Wikimedia Foundation, n.d. Web. 24 Oct. 2014. <http://en.wikipedia.org/wiki/Nash_equilibrium>.

Polak, Ben. "ECON 159 - Lecture 5 – Nash Equilibrium." Open Yale Courses. Yale, n.d. Web. 22 Oct. 2014. <http://oyc.yale.edu/economics/econ-159/lecture-17>.

17

Polak, Ben. "ECON 159 - Lecture 17 - Backward Induction: Ultimatums and Bargaining." Open Yale Courses. Yale, n.d. Web. 22 Oct. 2014. <http://oyc.yale.edu/economics/econ-159/lecture-17>.

Riley, Geoff. "Oligopoly & Game Theory." Tutor2u. Tutor2u, 23 Sept. 2012. Web. 23 Oct. 2014. <http://tutor2u.net/economics/revision-notes/a2-micro-oligopoly-game-theory.html>.

Simley, John. "Game Theory." Reference for Business - Encyclopedia of Business, 2nd Ed. Ed. Karl Hell. Advameg, Inc, n.d. Web. 23 Oct. 2014. <http://www.referenceforbusiness.com/encyclopedia/For-Gol/Game-Theory.html>.

Spaniel, William. "Game Theory 101: Backward Induction." Gametheory101. William Spaniel, n.d. Web. 24 Oct. 2014. <http://gametheory101.com/Backward_Induction.html>.

Spaniel, William. "Game Theory: The Ultimatum Game." Gametheory101. William Spaniel, n.d. Web. 24 Oct. 2014. <http://gametheory101.com/The_Ultimatum_Game.html>.

18