Prisoner s Dilemma. Veronica Ciocanel. February 25, 2013
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1 n-person February 25, 2013
2 n-person Table of contents 1 Equations 5.4, Types of dilemmas 4 n-person
3 n-person GRIM, GRIM, ALLD Useful to think of equations 5.4 and 5.6 in terms of cooperation and defection: Then: C D C R(CC) S(CD) D T (DC) P(DD) GRIM ALLD GRIM mr S+(m-1)P ALLD T+(m-1)P mp GRIM GRIM GRIM mr (m-1)r+s GRIM (m-1)r+t (m-1)r+p
4 n-person Regular matrices In the Reactive Strategies section, Nowak looks at the probability distribution x t after t rounds of game: x t+1 = x t M. x = xm gives a unique eigenvalue and normalized eigenvector if M is regular, i.e. k > 0 such that M k has all positive entries. Regular = power-positive + stochastic, used in structured population models.
5 n-person Fact that there is a unique eigenvalue and the entries of the associated eigenvector are all positive is due to the Perron-Frobenius Theorem. Easier way to check for a regular matrix Result by Horn and Johnson: If M has non-negative entries, then M is power-positive if and only if M n2 +n 2 has positive entries (where M is an n n matrix).
6 n-person Types of dilemmas Focus on zero-sum games, and pure game strategies; mixed strategies lead to larger probability spaces to consider. One way to solve a 2 2 payoff matrix would be to find its saddle, i.e. the combination of strategies in which each player can get highest possible payoff assuming best possible play by the opponent. Saddle point is entry a ij which is the minimum of row i, and maximum of column j. Example: 4-saddle point A B A 4 5 B 1 3
7 n-person Types of dilemmas From any introductory Game theory book, in zero-sum games, a ij is a saddle point iff outcome {i, j} is a pure strategy Nash equilibrium (connects Von Neumann s and Nash s theories). Reminder: Nash equilibrium is the combination of strategies in which no player has any benefit from changing strategies assuming that the opponent(s) don t change strategies. In non-zero sum games, there are equilibrium points that are not saddle points (not achieving best possible payoff). Example: prisoner s dilemma: C D C (3, 3) (1, 4) D (4, 1) (2, 2)
8 What if no saddle? Equations 5.4, 5.6 n-person Types of dilemmas General payoff matrix A B A a b B d c No saddle point means one of the two options: a > b, b < c, c > d and d < a (1) or a < b, b > c, c < d and d > a (2) Option (1): A and B are strict Nash. Is this a contradiction? Option (2): No strict Nash.
9 n-person Types of dilemmas Definition and classification of dilemmas Given the payoff matrix: C D C R(CC) S(CD) D T (DC) P(DD) The conditions for the game to be a dilemma are: always beneficial if the other cooperates: R > S (CC > CD) and T > P (DC > DD) sometimes beneficial to defect: T > R (DC > CC) or P > S (DD > CD) mutual cooperation better than mutual defection: R > P (CC > DD)
10 Types Equations 5.4, 5.6 n-person Types of dilemmas The only possible permutations of T, R, S, P: T > R > S > P, mutual defection worst, example: Chicken game, hawk-dove, snow-drift R > T > P > S, better to cooperate with cooperator, example: Stag hunt T > R > P > S, better to defect with cooperator, example:
11 n-person n-person prisoner s Dilemma: Tragedy of the Commons a a Andrew Moore, CMU Problem: You graze goats on the commons to eventually fatten up and sell. The more goats you graze the less fed up they are. And so the less money you make when you sell them. Facts and questions: Selling price per goat: 36 G How many goats would a rational farmer graze? What would he earn? How about a group of n farmers?
12 n-person n farmers i th farmer has an infinite space of strategies g i [0, 36] An outcome of (g 1, g 2, g 3,..., g n ) will pay to the i th farmer: n g i 36 j=1 g j
13 n-person Assume a pure Nash equilibrium exists: (gi, g 2,...g n ). Then gi = arg max gi [Payoff to farmer i, assuming the other players play g1,..., g i 1, g i+1,..., g n ] For convenience, can use notation G i = j i g j. Thus gi = arg max gi [g i 36 gi G i ], and g i must satisfy g i 36 g i G i g i = 0, = g i = (g g i 1 + g i g n ).
14 n-person Thus all equilibrium solutions are equal, call it gi = g. Then g = (n 1)g = g = 72 2n At the Nash equilibrium a rational farmer grazes 2n+1 goats. In total, there will be n n+1 = 36 2n+1 goats grazed (as n, 36 goats). 432 Each farmer has a profit of (= 1.26 cents if 24 (2n+1) 3/2 farmers). How much if all farmers would cooperate? 36 n (= 3.46 cents if 24 farmers)
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