INFORMAL INTRODUCTION TO GAME THEORY

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Game Theory deals with conflict situations. A conflict situation (game) is asituation in which two or more individuals (players) interact and thereby jointlydetermine the outcome. Each participating player can partially control the situa-tion, but no player has full control. They interact with one another, each one in aneffort to obtain that outcome that is most profitable for him.

The aim of game theory is to provide a solution (a characterization of rationalbehavior) for every game.

Traditionally, games have been divided into two classes: cooperative games andnoncooperative games. We restrict ourselves to noncooperative games. A nonco-operative game is played without any possibility of communication, correlation or(pre)commitment, except for those that are explicitly allowed by the rules of thegame. Hence, all relevant aspects of the game should be captured by the rules ofthe game.

A solution in a nooncooperative game is a set of recommendations, which telleach player how to behave. This solution should be consistent, i.e. no player shouldhave an incentive to deviate from his recommendation. Hence, a solution must beself-enforcing : As long as every player follows his recommendation, it should notbe in my interest to deviate.

R

1, 3

L

1

b

0, 0

a

2, 1

2

Figure 1. A perfect information game.

Example 1 (Backwards Induction). Consider the game in Figure 1. The rules ofthe game are as follows. The game starts with player 1 choosing between L and R.If player 1 chooses R the game ends, player 1 gets a payoff equal to 1 and player2 gets a payoff equal to 3. If player 1 chooses L the game continues and it is nowplayer 2’s turn to move. Player 2 can choose either a or b. If player 2 chooses a thegame ends, player 1 gets 2 and player 2 gets 1. If player 2 chooses b the game alsoends and both player 1 and player 2 get a payoff equal to 0.

Given such a game, is the strategy profile (R, b) (i.e. “player 1 moves R andplayer 2 would choose b in case he was called to move”) a good solution? It is,at least, a Nash equilibrium. No player can improve by unilaterally deviating fromsuch a description.

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Not Confess ConfessNot Confess 10, 10 0, 11

Confess 11, 0 3, 3

Figure 2. Prisoner’s Dilemma.

Player 1 plays R because he believes that if he played L player 2 would choose band he would get a payoff equal to 0 instead of 1 that he can guarantee by movingR. Player 2 plays b because, given that player 1 is playing R, whatever he playscannot change his payoff.

Should player 1 believe player 2’s threat? It does not look so. Player 2 willobviously move a whenever he has to move. Therefore, player 1 can anticipate thisand deviate to L. In other words, the only strategy profile that satisfies backwardsinduction is (L, a).

Example 2 (Prisoner’s Dilemma). Two suspects are partners in a major crimewho have been captured by the police. Each suspect is placed in a separate cell,and offered the opportunity to confess to the crime. If neither suspect confesses,they can still be convicted of a minor crime and they will spend a shot period oftime in prison (for whatever reason, this will give each of them a payoff equal to10). However, if one prisoner confesses and the other does not, the prisoner whoconfesses testifies against the other in exchange for going free (which gives him apayoff equal to 11), while the prisoner who did not confess goes to prison for a longterm (which gives him a payoff equal to 0). If both prisoners confess, then both aregiven a reduced term, but both are convicted, which we represent by giving each 3units of utility: better than having the other prisoner confess, but not so good asgoing free. The game is represented in Figure 2.

The most attractive strategy combination is (Not Confess ,Not Confess). How-ever, a sensible theory cannot prescribe this strategy profile as a solution. Supposeit is suggested to play (Not Confess ,Not Confess), each player has an incentive todisobey his recommendation as long as he expects his opponent to obey it.

Suppose prisoner 1 confesses. Prisoner 2 is better off confessing than not. Thesame is true if prisoner 1 does not confess. This reasoning implies that prisoner1 should confess. The argument that shows that prisoner 2 should also confess isanalogous. Game Theory has to prescribe (Confess ,Confess) as the solution in anoncooperative context.

RL

1

b

0, 0

a

2, 1

b

1, 3

a

1, 3

2 a b

L 2, 1 0, 0R 1, 3 1, 3

Figure 3. A simultaneous move game derived from the game in Figure 1.

Example 3 (Admissible Equilibria). Consider now the game in Figure 3. Thefigure offers an extensive form and a normal form representation of the same game.

INFORMAL INTRODUCTION TO GAME THEORY 3

The rules of the game are as follows. Player 1 has to choose between L and R, andplayer 2 has to choose between a and b. Each of them has to make a choice withoutknowing the other player’s choice. Payoffs are as described in the figure.

The game is a modification of the game in Figure 1. Player 2 is now calledto move regardless of player 1’s choice, but his choice is only relevant in terms ofpayoffs if player 1 plays L.

Is now (R, b) a good solution of the game?Let us pose the question in different terms. Do the games in Figure 1 and Figure 3

represent the same strategic situation? Our answer must be yes. In either case,player 1 and player 2 know that player 2’s strategy choice is only relevant in caseplayer 1 chooses L. Consequently, the strategy profile (R, b) cannot be considereda good set of recommendations.

More generally, a transformation of the game tree like the one applied in thecurrent example is irrelevant for correct decision making. The transformed tree ismerely a different presentation of the same decision problem, and decision theoryshould not be misled by presentation effects. Otherwise we would be saying thatdecision theory and game theory are useless in real-life applications, where problemspresent themselves without a specific formalism such as a tree.

But the argument that we used in Figure 1 to discredit (R, b) cannot be used inFigure 3 because when player 2 has to move he does not know whether player 1 hasplayed R or L. If our previous reasoning is valid then there must be an argumentthat applied solely to the game as represented in Figure 3 would discredit thestrategy profile (R, b). Such an argument is the following: Why would player 2play strategy b? After all strategy b is never strictly better than strategy a, andsometimes (when player 1 plays L) is strictly worse. We say that strategy b is(weakly) dominated by strategy a.

In any case, notice that, as in the game of Figure 1, (R, b) is a Nash equilibrium.Also notice that the strategy profile (L, a) is a Nash equilibrium too, but in thisequilibrium all players are using admissible (i.e. not dominated) strategies.

Appendix A. Tutorial Questions

Exercise 1. Write down the normal form representation of the following gamesand find all Nash equilibria, in pure and mixed strategies.

The Battle of the Sexes Husband and wife agree to meet this evening, but cannot recall if theywill be attending the opera or a boxing match. He prefers the boxingmatch and she prefers the opera, though both prefer being together tobeing apart. They are in separated locations and cannot communicate byany means. Each of them has to decide where to go. Suppose that meetingtheir partners in their favorite event pays 3, while meeting their partnersto the other event pays 1. If they do not meet they both get a payoff equalto 0.

Matching Pennies Two children first select who will be represented by “same” and who will berepresented by “different”. Then, each child conceals in his palm a pennyeither with its face up or face down. Both coins are revealed simultaneously.If they match (both are heads or both are tails), the child “same” wins. Ifthey are different (one heads and one tails), “different” wins. Let us assumethat the winner gets a payoff equal to 1 and the loser a payoff equal to −1.

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The Game of Chicken Two people take their cars to opposite ends of a narrow road and startto drive toward each other. The one who swerves to prevent collision isthe “chicken” and the one who keeps going straight is the winner. Let thepayoff to the “chicken” be −1 and the payoff to the winner 1. If they bothswerve they get a payoff equal to 0. If nobody swerves they both get apayoff equal to −2.

The Stag Hunt Game Each of two hunters has two options: he may remain attentive to the pursuitof a stag, or he may catch a rabbit. If both hunters pursue the stag, theycatch it and share it equally, giving each a payoff equal to 2. If one of thehunters devotes his energy to catching a rabbit, he catches the rabbit butthe stag escapes. Catching the rabbit gives a payoff equal to 1 and catchingno prey gives a payoff equal to 0.

Rock, Paper, Scissors Two children simultaneously make one of three symbols with their fists -a rock, paper, or scissors. Simple rules of “rock breaks scissors, scissorscut paper, and paper covers rock” dictate which symbol beats the other. Ifboth symbols are the same, the game is a tie. Similar to the two-strategyMatching Pennies game. Let us assume that the winner gets a payoff equalto 1 and the loser a payoff equal to −1. If they tie they both get a payoffequal to 0.