Chapter 2: Commitment

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1 Chapter 2: Commitment Outline A. Modular rationality (the Gianni Schicchi test). Its conflict with commitment. B. Puzzle: our behaviour in the ultimatum game (more generally: our norms of fairness) violate modular rationality Solution: evolutionary game theory shows how behaviour violating modular rationality could evolve The trembling hand: neither recombination nor mutation guarantees modular rationality C. Theology of commitment: Gauthier and McClennen give up modular rationality, but Skyrms rejects this idea. D. Summary: evolution and modularity

2 A. Modular Rationality A plan or policy or strategy (or even a disposition) is a rule that prescribes a choice for an agent at each of a sequence of choice points. Some plans (especially threats, promises) entail individual choices that would be irrational to carry out if called upon to do so. Ex: All-out nuclear retaliation (M.A.D.) Ex: Exposing Gianni Schicchi Ex: Mass resignation (nurses strike) Def: A plan involving a sequence of choices exhibits modular rationality if it specifies a rational choice at each point, relative to the choice situation at that point. Folk wisdom: Modular rationality is a necessary requirement for any plan to be credible. But only credible threats can achieve their desired effect. Remarks: 1) Rationale for necessity claim: what reason for action other than max. utility? 2) Establishing a precedent may provide utility sufficient to preserve the rationality of carrying out otherwise unattractive choices entailed by a plan. 3) Not a sufficient condition for a plan to be rational: martingales. 4) Best option may be to eliminate the later choice point (e.g., Doomsday Device)

3 B. Actual Practice 1. The Ultimatum Game $100 to be divided between two players. Player 1: proposes a division (x, 100-x) where 0 x 100 Player 2: The puzzle: 2. Solutions - accept for payoff of (x, 100-x) OR - reject for payoff of (0, 0) The profiles (Propose (x, 100-x), Accept if at least 100-x) are all Nash equilibria. Suppose Player 2 gets to make a threat: I ll reject any offer less than 100 x. Assuming money has positive utility, every such threat appears to violate the Gianni Schicchi test, except where 100 x is 1 cent. Hence, no such threat is credible. Yet in reality, most often player 1 offers a nearly even split, and player 2 rejects a greedy offer. Why? 1) Within the rational choice paradigm: people are maximizing utility rather than EMV, and utility includes a value attached to fairness. Objection: But where do the norms of fairness come from? Should they not have been eliminated by evolution? 2) Evolutionary model. Simplified case: 10 pieces of cake. Player 1: demand 9; demand 5 Player 2: accept all; accept none; accept 9 but not 5; accept 5 but not 9 If we assume a single population, a combined strategy must tell you what to do if you are player 1, and what to do if you are player 2.

4 8 options: If player 1 If player 2 S1: Gamesman Demand 9 Accept all S2 Demand 9 Reject all S3 Demand 9 Accept 5, reject 9 S4: Mad Dog Demand 9 Accept 9, reject 5 S5: Easy Rider Demand 5 Accept all S6 Demand 5 Reject all S7: Fairman Demand 5 Accept 5, reject 9 S8 Demand 5 Accept 9, reject 5 Assume random pairings, and randomly assigned role of player 1/player 2. 1) Normative approach. Compute expected payoffs. S1 dominates S2- S4, and S5 dominates S6-S8. And in the remaining 2x2 table, S1 dominates S5. So rational choice approach prescribes S1. 2) Evolutionary approach. Computing equilibria: (ps4, (1-p)S7): p = 0, ½, 1 (Possible values) (ps3, (1-p)S8): p = 0,.7, 1 (Possible values) Interesting cases: From equal starting proportions for all 8 strategies, evolves to (87% S1, 13% S4). (Note that S1 dominates S4, but only does better against the strategies S5 through S8 that are dying out.) If we start with 30% S7 and rest divided equally, evolves to (64% S7, 36% S5). (S5 beats S7 versus S1 - S4, but does equally once these greedy strategies are out.) Remarks: the weakly dominated strategies need not be eliminated. The key is the rate of extinction of the strategies where the free rider does better. Example: Fairman is not modular rational, yet it can survive.

5 3. Trembling Hand Idea: Take into account wobbles mistakes that lead you or opponent to choose a dominated strategy. An equilibrium is (trembling hand) perfect if it remains an equilibrium even when all dominated choices are given a small positive probability of occurring. Theorem: Any strategic equilibrium is modular rational if and only if it is trembling hand perfect. Example: In a population consisting entirely of Fairmen and Easy Riders, both do equally well. But if we allow for the small probability of somebody getting greedy and demanding 9, then Easy Rider does better. Thus, Fairman is ruled out. Evolutionary analogue: mutation and recombination might allow new strategies to emerge. Question: Do these phenomena rule out evolutionary equilibria with Fairmen? a) Recombination i) Splicing Each of our strategies has three parts : - what to demand if player 1 - whether or not to accept 9 if player 2 - whether or not to accept 5 if player 2 Recombination can produce any strategy if the parts are available among existing ones in the population. ii) Evolutionary trembling hand An evolutionary equilibrium is rejected if it would collapse under recombinations. Ex: S4 (Mad Dog), S7 (Fairman). Unstable Equilibrium: 0.5, 0.5. S4: Demand 9; Accept 9; Reject 5 expect 5 vs. S4, 0 vs. S7: avg. 2.5 S7: Demand 5; Accept 5; Reject 9 expect 5 vs. S7, 0 vs. S4: avg. 2.5 Recombination from these two can produce any strategy. But against a population consisting of almost 50/50 S4 and S7, Gamesman expects avg while Easy Rider expects avg Either will start to emerge. Ex: Any Fairman/Easy Rider equilibrium, or Mad Dog/Gamesman equilibrium. No new strategies can emerge under recombination, so consistent with evolutionary trembling hand.

6 b) Mutation Transformation: a change in a portion of a strategy (e.g., Demand 9 Demand 5) Every transformation has some probability, so that every strategy will pop up in the long run. Claim: Although it seems that Dominant strategies would take over, this may not happen because there could be other mutations that counterbalance. More specifically: Gamesman/Mad Dog equilibria and Fairman/Easy Rider equilibria can persist even with some types of mutation.

7 C. Theology of Commitment We know that classical game theory rules out behaviour that is not modular rational: commitments that would entail choosing dominated options at some choice point. Evolutionary game theory provides one explanation for why such behaviour is in fact found. Alternative approach (Gauthier, McClennan): Commitment is rational. Give up modular rationality in favour of (e.g.) constrained maximization. 1. Consistency argument: It would be inconsistent to prefer a disposition, but then to judge it not rational to carry out the entailed action. Objection: No inconsistency because the judgements are made at different times and with different information. Ex: Doomsday machine. Rational to build it in belief that it will deter a nuclear attack. Rational not to launch retaliation (should the machine malfunction) in the event of an attack: deterrence failed. Contra Gauthier, it can be rational to want a disposition without every choice in accordance with that disposition counting as rational. 2. Ulterior motive: if rationality meant joint choice of an optimal joint strategy, plus commitment to that strategy, then we could derive rule utilitarianism from rationality alone. Objections: (1) Best theories of human rationality suggest that an ideal agent maximizes subjective expected utility: hence, makes modular rational choices. (2) Dark side of commitment: commitments can lead to disastrous choices (Tit-for-tat, feuds, etc.) (3) Alternative explanation for committed behaviour: build sympathy, justice, desire to maintain character into our utility.

8 D. Modularity and Evolution Main result of chapter: Committed behaviour that fails test of modular rationality can evolve and be part of an evolutionary equilibrium.

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