Analytic Preliminaries for Social Acceptability of John Asker February 7, 2013 Yale Law School
Roadmap Build on case studies of environments where extra-legal norms are at least as important as formal legal rules and standards. Develop analytic tools that are useful for putting structure on these observations Game Theory Repeated games and the folks theorems The tools will be drawn from economic theory. At least in my view, the point of a theoretical framework is to organize observation so that we can make meaningful inference that maps back into conceptual development and policy. Understanding what phenomena are interesting and informative Assessing the appropriateness of frameworks for answering questions of interest Measurement of magnitudes of interest Theory, observation and measurement go hand-in-hand.
Case Studies Ellickson, Robert (1989), A Hypothesis of Wealth-Maximizing Norms: Evidence from the Whaling Industry Journal of Law, Economics and Organization, 5(1), 83-97 Property Law Grief, Avner (1993), Contract Enforceability and Economic Institutions in Early Trade: The Maghribi Traders Coalition American Economic Review, 83(3), 525-548. Contract Law / Law of Agency
Case Studies: Ellickson, Robert (1989), A Hypothesis of Wealth-Maximizing Norms: Evidence from the Whaling Industry Journal of Law, Economics and Organization, 5(1), 83-97 Issue: Two different approaches to establishing property rights over whales hunted by multiple ships in international waters. Why these approaches (vs. Others)? Why one approach in one market and one approach in another?? Adoption by courts
Case Studies: Greif, Avner (1993), Contract Enforceability and Economic Institutions in Early Trade: The Maghribi Traders Coalition American Economic Review, 83(3), 525-548. Issue: Diamond wholesale market uses extensive arbitration internal to the community that severely penalizes taking disputes to courts Why this approaches? How does it work? How is it enforced? Capable of being supported by the court system? Why supplanted by the Merchant's Law? see the Milgrom et al paper.
Analytic Tools Objective: Acquire tools to develop structures to organize and interrogate this sort of descriptive empiricism. Review of Game Theory Repeated Games (enforcement of norms) Basic Folk Theorems (which norms can be enforced) (if time) Revisit case studies and point out where the framework is helpful and deficient A useful reference for applications of Game Theory is Gibbons (1992) Game Theory for Applied Economists (about $40 on amazon.com)
Review of Game Theory Tools I am going to assume you have seen this material in some form before. I am going to avoid mathematical formalism, for the most part.
Review of Game Theory Tools What are the elements of a game? Players Rules Strategies Payoffs Example: Normal Form Game Player B L C R 5 6 7 T 9 8 1 3 5 6 Player A M 1 2 0 7 6 8 B 2 3 0 Moves are simultaneous. Which moves make sense?
Review of Game Theory Tools Solution Concepts I: Strict Dominance Strictly Dominant strategy: payoff is strictly greater than any other strategy regardless of other players strategies. Strictly Dominated strategy: payoff is strictly lower than some other strategy regardless of other players strategies. An equilibrium in strictly dominant strategies is a set of strategies, one for each player such that each strategy is strictly dominant. Example: Normal Form Game Player B L C R 5 6 7 T 9 8 1 3 5 6 Player A M 1 2 0 7 6 8 B 2 3 0
Review of Game Theory Tools Solution Concepts II: Weak Dominance Weakly Dominant Strategy: payoff is weakly greater than (i.e. at least as good as) any other strategy regardless of rival s choice, at strictly greater for at least one strategy profile available to other players. An equilibrium in weakly dominant strategies is a set of strategies, one for each player such that each strategy is weakly dominant. Example: Normal Form Game Player B L C R 5 6 7 T 9 3 1 3 5 6 Player A M 1 2 0 7 6 8 B 2 3 0
Review of Game Theory Tools Solution Concepts III: Nash Equilibrium Best response set: Given strategies played by other players, the optimal strategies for a player. This need not be a singleton. Nash Equilibrium: A set of strategies, one for each player, such that each player is playing a best response. Example: Normal Form Game Player B L C R 5 6 7 T 0 3 1 3 5 3 Player A M 1 2 0 2 6 8 B 2 3 0
Review of Game Theory Tools Canonical Game Forms Prisoners Dilemma: Collectively optimal outcome is different from what is individually rational C Player B D C 10 15 10-3 -3 0 Player A D 15 0
Review of Game Theory Tools Canonical Game Forms Prisoners Dilemma: Collectively optimal outcome is different from what is individually rational Player B H M L 10 15 12 H 10 3 2 3 5 7 Player A M 15 5 2 2 2 3 L 12 7 3
Review of Game Theory Tools Canonical Game Forms Coordination Game: Equilibrium outcomes rely on coordination occurring successfully. Think about which side of the road to drive on C Player B D C 10 0 10 0 0 10 Player A D 0 10
Games where moves occur over time Nash Equilibrium: A set of strategies, one for each player, such that each player is playing a best response. Subgame Perfect Nash Equilibrium: A Nash Equilibrium that every agent is playing optimally at every decision node. More formally, play within any subgame is also a Nash Equilibrium.
Player 1 Games where moves occur over time: Bully Leave alone Example: Player 2 5 NE vs. SPNE Hand over lunch Fight 10 10 0 0-3
Player 1 Games where moves occur over time: Bully Leave alone Example: Player 2 5 NE vs. SPNE Hand over lunch Fight 10 10 0 0-3 NE: 1. <LA,F> 2. <B,HOL> SPNE: 1. <B,HOL>
Repeated Games: outside legal sanctions Basic Insight: If parties interact repeatedly, use future as source of leverage: threaten to punish rival if it behaves non-cooperatively. Recall the following from our case studies: Whaling: There is no finer example in history of communal enterprise than the Nantucket Whale Fishery. The inhabitants were uniquely situated for united effort Through intermarriage they were generally related to one another, and in fact were more like a large family than a civic community Diamonds: Sugden developed a model of exchange that demonstrates how, under certain conditions, a market norm that normally results in cooperation can be a stable, though not unique, equilibrium even when there appear to be incentives for individuals to be free riders and transactors occasionally make mistakes (breach unintentionally). The game is an adaptation of the classic prisoner's dilemma model in which the following conditions hold
Repeated Games: outside legal sanctions Stage game: Prisoners Dilemma: Collectively optimal outcome is different from what is individually rational C Player B D C 10 15 10-3 -3 1 Player A D 15 1 Define π(c,c), π(d,c), π(d,d), π(c,d) In this example these are : 10, 15, 1 and -3 respectively.
Repeated Games: outside legal sanctions Game Form: Play the stage game repeatedly, off into some infinite horizon Discount future using discount factor δ Recall that the value of a stream of payments, every period from today into an infinite future is 1/(1-δ) [I will do this on a whiteboard, if you are not familiar with it] Definition of a Strategy: A strategy is a complete contingent plan of action.
Repeated Games: outside legal sanctions Claim: in the repeated game, the cooperative outcome (C,C) can be a (Subgame Perfect) Nash equilibrium. Equilibrium strategies that generate this: If all players play C in the past, stick to it. If one player has ever played D, play D. (This punishment is what supports the equilibrium.) This is known a Grim Trigger Strategy. This equilibrium is not unique. If everyone else plays this strategy, what is the incentive to deviate? π(c,c) /(1-δ) ; versus π(d,c) + δπ(d,d) /(1-δ) Thus the necessary and sufficient condition for this to exist is that π(c,c) /(1-δ) > π(d,c) + δπ(d,d) /(1-δ)
Repeated Games: outside legal sanctions Observations: Comparative Statics Empirical Relevance Next: Multiplicity: The Folk Theorem
Repeated Games: The Folk Theorem Basic Intuition and mechanics: The idea here is that payoffs which are not as good as the socially optimum, but better than the static Nash outcome, can also be supported essentially the mechanics are much the same but showing it is a little more subtle.
Repeated Games: The Folk Theorem Feasible payoffs: Payoffs are feasible if they are a weighted average of the payoffs in the normal form game. In the example, feasible payoffs are shaded. -3,15 Player A C D Player B C D 10 15 10-3 -3 1 15 1 Definitions 10,10 1,1 15,-3
Repeated Games: The Folk Theorem Average payoff The average payoff of a stream of payments is: (1-δ)Σ t δ t-1 π t Definitions
Repeated Games: The Folk Theorem Theorem (a folk theorem): Let G be a finite, static game of complete information. Let (e 1,.,e n ) denote the pay-off to each of n players, from a Nash equilibrium of G, and let (x 1,,x n ) denote any feasible payoff profile from G. If x i >e i for every player and if δ is sufficiently close to one, then there exists a subgame-perfect nash equilibrium of the infinitely repeated game that acheives (x 1,,x n ) as an average payoff.
Repeated Games: Theorem (a folk theorem): Let G be a finite, static game of complete information. Let (e 1,.,e n ) denote the pay-off to each of n players, from a Nash equilibrium of G, and let (x 1,,x n ) denote any feasible payoff profile from G. If x i >e i for every player and if δ is sufficiently close to one, then there exists a subgame-perfect nash equilibrium of the infinitely repeated game that achieves (x 1,,x n ) as an average payoff. The Folk Theorem; -3,15 Player B C D What it means 10,10 Player A C 10 15 10-3 -3 1 D 15 1 1,1 15,-3 These average payoffs can be supported
Repeated Games: The Folk Theorem Sketch of proof Theorem (a folk theorem): Let G be a finite, static game of complete information. Let (e 1,.,e n ) denote the pay-off to each of n players, from a Nash equilibrium of G, and let (x 1,,x n ) denote any feasible payoff profile from G. If x i >e i for every player and if δ is sufficiently close to one, then there exists a subgame-perfect nash equilibrium of the infinitely repeated game that acheives (x 1,,x n ) as an average payoff. The idea is to use a public randomizing device to generate the average payoffs (i.e. a dice decides which action each person plays at the start of a round, and everyone can see this dice roll). Then everyone, does what they are told. Deviations are punished using the grim trigger strategy as worked through before. It s a tiny bit more annoying in terms of notation, but the math is the same. See Gibbons, p100 101. What is the observational analog of this? What is the empirical relevance?
Repeated Games: Multiplicity, even on the efficient frontier Exercise: Coordinating duopolists Two firms, different marginal costs, who try to coordinate on the monopoly price π 1 (c,c) /(1-δ) > π 1 (d,c) + δπ 1 (d,d) /(1-δ) π 2 (c,c) /(1-δ) > π 2 (d,c) + δπ 2 (d,d) /(1-δ) We will do the boardwork: the point is that there is a range of divisions of the market that are both supportable using grim triggers, that also maximise the joint wealth of our two firms The question, is then on what basis do they choose the way to split the wealth?
Implications: Useful insights and unresolved issues Where do norms come from (how are equilibria selected?). How poor can enforcement be? How bad can monitoring be? Does this ignore sociological motivations? (i.e. intrinsic value from a sense of belonging or conformity?) is this observationally distinct? What does the legal system add to this way of coordinating? (Is there a sense in which this is merely the Coase theorem re-expressed?)