The Limits of Reciprocal Altruism Larry Blume & Klaus Ritzberger Cornell University & IHS & The Santa Fe Institute
Introduction Why bats? Gerald Wilkinson, Reciprocal food sharing in the vampire bat. Nature 308, 1984: 181 184., Food sharing in vampire bats. Scientific American 262, 1990: 76-82. Sustaining cooperation? David Kreps, Paul Milgrom, John Roberts and Robert Wilson, Rational cooperation in the finitely repeated prisoners dilemma. JET 27, 1982. Robert Axelrod, The Evolution of Cooperation, 1984. Elinor Ostrom, Governing the Commons: The Evolution of Institutions for Collective Action, 1990.
Social Norms What Are Social Norms? Norms are socially shared prescriptions for behavior. They emerge from behavioral regularity. What Sustains Social Norms? Social Expectation Lessig 1995, Acemoglu and Jackson 2013 Regulation of Payoffs Institutions Reciprocal altruism.
Limits to Reciprocal Altruism The survival of reciprocal altruism in repeated games is a consequence of the folk theorem, that if the future is sufficiently immediate, the potential for manipulating future outcomes can discipline current choices. BUT Experimenting is costly. The short-term costs of learning to cooperate can destroy the long-term stability of cooperation. The benefits to cooperation cannot always be reciprocated. Folk theorems require that the only way the present affects the future is through what individuals believe about the future. Is reciprocal altruism sustainable when the present physically constrains the future? Suppose there are state variables other than beliefs.
Model States Two individuals each eat 1 per day Each individual draws a random endowment once a day. The endowment is 2 with pr p, and 0 otherwise. Individuals maintain an inventory. A individual may share 1 with the other individual. A individual dies when she fails to eat. The inventory space is I, which describes each individual s current inventory. b i = 1 is death, an absorbing state.
Model Payoffs Let Ω denote the sample space on which the processes are built: ω t = (ω 1t, ω 2t ) where each ω it describes the outcome of i s date-t endowment draw, success or failure. From each strategy profile (σ 1, σ 2 ) and the initial inventory b = (b 10, b 20 ) compute τ i b (ω) = inf{t : b it = 1}, the time at which i dies. Individual i s payoff function is u i (σ 1, σ 2 ) = τ i b (ω) t=0 δ t.
Dynamics I Irreversible Accumulation An action for individual i is a choice to share S or withhold W 1 unit from individual j. A = {S, W}. A state q Q of the game is a quadruple q = (b 1, b 2, ω 1, ω 2 ) where (b 1, b 2 ) I are the individual s inventory levels, and ω i is describes the outcome of individual i s endowment draw. State q t describes the date-t physical situation after endowments are realized. Taking s = S = 1 and f = W = 0, the dynamics are b 1t b 1t, ω 1t b 2t b 2t, ω 2t b 1t + ω 1t a 1t + a 2t 1 b 2t + ω 2t a 2t + a 1t 1 b t q t b t+1
Strategies A norm is a map n : I A. N denotes the set of norms. A partial history is a structure (q 0, a 10, a 20,..., q t, a 1t, a 2t ). Let H denote the set of partial histories. h t H is the sequence of states and actions through date t. A strategy for individual i is a map σ : H N where σ(h t 1 ) is the norm employed at date t.
Some Norms Autarchy. Each individual contributes to and withdraws from her own inventory. No sharing takes place. Simple Sharing. A successful individual shares with an unsuccessful individual. Wealth-Based Sharing. A successful individual shares with the other individual iff she is wealthier. Full Sharing. Split 50-50.
Questions The possibilities for cooperation: Is sharing over some or all of the state space optimal? Yes, on all of I. For large δ, are there equilibria which support sharing on some or all of the state space? Depends on p, and only for some part of I. For large δ, are there equilibria which achieve the welfare optima? No.
Folk Theorems These questions are normally answered by folk theorems. Requirements for the folk theorem: The set of feasible long-run average payoffs is state-independent. The long-run average min-max payoffs are state-independent. The dimension of the set of long-run average feasible payoffs is 2. In our game. For p < 1/2 the only feasible long-run average payoff is 0. For p > 1/2, the maximal long-run average payoff is state-dependent. Prajit Dutta, A folk theorem for stochastic games JET 66, 1995.
Value Functions Strategies determine a stochastic process on I. τ i b is the first time the process hits 1. The value of being at b I is τ i b (ω) V i (b) = E δ t lim V i (b) = δ 1 t=0 = 1 δ E { δ τi b (ω)} 1 δ 1 + { E τ i b (ω)} { if pr τ i (ω) b < } = 1, otherwise The average discounted value at b I is ADV i (b) = 1 δ E { δ τi b (ω)}, lim ADV i (b) = { pr τ i b (ω) = }. δ 1
Results Optimal Norms The welfare-optimal norm is full sharing. The welfare-optimal norm for sharing out of new endowment is wealth-based sharing.
Results Achieving Optima We look for SPE with a grim trigger. A successful individual may defect from sharing and revert to autarchy but with one more unit. The social optima are never equilibria. In both games there is no δ < 1 such that a sharing norm is supported at (0, 0). In the flow game, there is no δ < 1 such that sharing is supported when either individual has stock 0. Autarky is always an equilibrium. Autarky is the only equilibrium for p < 1/2.
Results Sharing Norms For any p > 1/2 there is a δ sufficiently large, simple sharing at every interior state is an equilibrium. For any p > 1/2 and for all sufficiently large δ there is an M such that wealth-based sharing at all states such that max{b 1, b 2 } min{b 1, b 2 } + 2M and autarky otherwise is an equilibrium. For any p > 1/2 there is no δ < 1 such that wealth-based sharing is an equilibrium at all interior states. For any p > 1/2 and for all sufficiently large δ ex ante full sharing is optimal everywhere except at (0, 0).