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Operations Research - Game Theory
Egwald's popular web pages are provided without cost to users. Game Theory - Introduction | Battle of the Sexes | Prisoner's Dilemma | Free Rider | Game of Chicken | Play a Game Online | Metagame Example Nonnegotiable Two Person Generalized (Non-Zero) Sum Games The Battle of the Sexes game was solved through negotiation backed by threats. The implementation of the negotiated settlement, going to the theatre, restaurant, and/or baseball game, was easy to monitor. That is, neither player could cheat because they had to execute their agreement by "doing it together." However, many interpersonal contexts exist where a negotiated settlement cannot be enforced. Each player has an incentive to break the agreement determined prior to the play of the game. However, both players could be better off if they stick to their agreement. In games like the Prisoner's Dilemma, the players face a conundrum. Individual rationality suggests a player cheat and break the agreement. Group rationality suggests they cooperate, even though neither player knows for sure that the other player will play as agreed. In fact, each player knows the other player has an incentive to cheat. If both players cheat, each player is much worse off than in the cooperative outcome. Hardheaded self-interest suggests each player make a self-defeating choice. The philosopher Max Black (117) calls this an entrapment situation. Such situations have been identified in many interpersonal, inter-organizational, and international contexts. The Trenchcoat Robbers To see this interplay of individual and group rationality more clearly, consider the following game tableau, with payoffs denoted in the utility of the outcome to each player, inspired by the Annals of Crime article by Alex Kotlowitz, "The Trenchcoat Robbers," from The New Yorker. Ray and Bill have been robbing banks for twenty years. They decide to rob one more bank before they "retire." They rob a bank, obtaining a huge sum of money, and successfully stash the loot to be retrieved when the heat is off. On their way home, the Police stop their car for a minor speeding infraction. Ray and Bill's descriptions fit that of the bank robbers, so the Police search their car, finding two unregistered firearms. The Police arrest Ray and Bill. Now, the Police interrogate Ray and Bill in separate rooms. In the payoff matrix below, the first (second) entry in each cell of the first row (column) represents the payoff to Bill (Ray) of confessing. The first (second) entry in each cell of the second row (column) represents the payoff to Bill (Ray) of not confessing. These payoffs are, of course, conditional on the choice of the other player.
If Ray and Bill both "keep mum" and do not confess to the crime — they will both go free for a utility of 4 units each. If Bill confesses and Ray does not — Ray goes to jail for 10 years, a utility of 0, but Bill, after just 1 year in jail, has the opportunity to spend his share of the money obtained in previous robberies plus the money they have not yet divided, a utility of 6. The situation is symmetrical if Ray confesses and Bill does not. If both confess, they spend 5 years in jail before either can enjoy his split of the proceeds from their robberies — yielding a utility of 2 each. X1 is the probability that Bill confesses; X2 is the probability that Bill "keeps mum;" X2 = 1 - X1; Y1 is the probability that Ray confesses; Y2 is the probability that Ray "keeps mum;" Y2 = 1 - Y1. Ray and Bill are stuck in a position that leaves each only a choice between evils — a dilemma. Having been arrested, they cannot extricate themselves from the entrapment situation. Thinking of Bill's choices, Ray thinks: 1. If Bill confesses — I should confess, because 2 is greater than 0. 2. If Bill "keeps mum" — I should confess, because 6 is greater than 4. The dominant strategy is (confess, confess) as dictated by individual rationality. It is an equilibrium strategy because any attempt by a player to switch strategies will leave the player worse off. Competent bank robbers, Ray and Bill designed many schemes to avoid the dilemma. Aware of the dilemma, they may have tried to change the rules of the game — to build trust between them — like going fishing together. If the schemes are successful, the dilemma does not exist, and the tableau does not represent their choices. Ray and Bill might have agreed in advance to confess if caught. Why? The Police's evidence as to their guilt cannot be conclusive. Why are they offering the confessor a deal? So, the Police's evidence is at best persuasive. Knowing this, why would either player not confess, if this were their only crime? Why trust a partner with whom you never worked before, and with whom you will never work again? The Police (and their social scientists) neither know the game tableau exactly, nor are aware of Ray and Bill's previous crimes. Not confessing indicates to the Police that other payoffs exist down the road, encouraging them to expand their investigation of Ray and Bill. So Ray and Bill confess or "keep mum," depending on their perception of the Police's case against them. Like their "Trenchcoat Robbers" counterparts, Ray and Bill have robbed many banks, with no inclination to turn themselves in. If the Police's evidence is conclusive enough, they should confess. The General Case In practice, the values of the payoffs in the Game Tableau could vary from one situation to another: Is this Ray and Bill's first bank job or not? The following Game Tableau uses letters to represent variables that can take on different values as payoffs.
For the Game Tableau to represent a Prisoner's Dilemma situation, the variables a, b, c, and d must obey the following restrictions: These restrictions represent the order of preferences for Ray and Bill. Hardheaded realism dictated that both Ray and Bill confess. The loss to either player of the "Keep Mum" pure strategy can be measured by the value of (a - c), the difference in the payoffs from the "Confess" and "Keep Mum" choices if the other player chooses "Confess." In the first Game Tableau, (a - c) = (2 - 0) = 2. If Ray and Bill expect to be in this situation repeatedly, and if they can monitor the other player's choice after each instance of a play of the game, small values of (a - c) might induce (Ray, Bill) = (Keep Mum, Keep Mum) outcomes, i.e. the group rationality outcome. Conversely, if the gain from double crossing the partner, (b - d) = (6 - 4) = 2, is large, the (Ray, Bill) = (Confess, Confess) outcomes are more likely to obtain, even if Ray and Bill made a committment to play the pure "Keep Mum" strategy. Black (126) has considered the notion of "committment," and Rapoport, the effect of "learning in games of repeated play."
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