Which way to cooperate

Cooperation in real-world dilemmas takes many forms. We introduce a class of two-player games that permits two distinct ways to cooperate in the repeated game. One way to cooperate is to play cutoff strategies, which rely solely on a player's private value to defection. The second cooperative strategy is to take turns, which relies on publicly available information. Our initial experiments reveal that almost all cooperators adopt cutoff strategies. However, follow-up experiments in which the distribution of values to defection are made more similar show that all cooperators now take turns. Our results offer insight into what form a cooperative norm will take: for mundane tasks or where individuals otherwise have similar payoffs, taking turns is likely; for difficult tasks that differentiate individuals by skill or by preferences, cutoff cooperation will emerge.experimental economics; cooperation; incomplete information; alternating; cutoff strategies; random payoffs

Preemptive Entry in Differentiated Product Markets

Models of spatial competition are typically static, and exhibit multiple free-entry equilibria. Incumbent firms can earn rents in equilibrium because any potential entrant expects a significantly lower market share (since it must fit into a niche between incumbent firms) along with fiercer price competition. Previous research has usually concentrated on the zero-profit equilibrium, at which there is normally excessive entry, and so an entry tax would improve the allocation of resources. At the other extreme, the equilibrium with the greatest rent per firm normally entails insufficient entry, so an entry subsidy should be prescribed. A model of sequential firm entry (with an exogenous order of moves) resolves the multiplicity problem but raises a new difficulty: firms that enter earlier can expect higher spatial rents, and so firms prefer to be earlier in the entry order. This tension disappears when firms can compete for entry positions. We therefore suppose that firms can commit capital early to the market in order to lay claim to a particular location. This temporal competition dissipates spatial rents in equilibrium and justifies the sequential move structure. However, the policy implications are quite different once time is introduced. An atemporal analysis of the sequential entry process would prescribe an entry subsidy, but once proper account is taken of the entry dynamics, a tax may be preferable.Product differentiation, rent dissipation, entry deterrence, entry timing, sequential entry

Unique bid auctions: Equilibrium solutions and experimental evidence

Two types of auction were introduced on the Internet a few years ago and have rapidly been gaining widespread popularity. In both auctions, players compete for an exogenously determined prize by independently choosing an integer in some finite and common strategy space specified by the auctioneer. In the unique lowest (highest) bid auction, the winner of the prize is the player who submits the lowest (highest) bid, provided that it is unique. We construct the symmetric mixed-strategy equilibrium solutions to the two auctions, and then test them in a sequence of experiments that vary the number of bidders and size of the strategy space. Our results show that the aggregate bids, but only a minority of the individual bidders, are accounted for quite accurately by the equilibrium solutions.

Testing Game Theory in the Field: Swedish LUPI Lottery Games

Game theory is usually difficult to test precisely in the field because predictions typically depend sensitively on features that are not controlled or observed. We conduct one such test using field data from the Swedish lowest unique positive integer (LUPI) game. In the LUPI game, players pick positive integers and whoever chose the lowest unique number wins a fixed prize. Theoretical equilibrium predictions are derived assuming Poisson- distributed uncertainty about the number of players, and tested using both field and laboratory data. The field and lab data show similar patterns. Despite various deviations from equilibrium, there is a surprising degree of convergence toward equilibrium. Some of the deviations from equilibrium can be rationalized by a cognitive hierarchy model

Self-tuning experience weighted attraction learning in games

Self-tuning experience weighted attraction (EWA) is a one-parameter theory of learning in games. It addresses a criticism that an earlier model (EWA) has too many parameters, by fixing some parameters at plausible values and replacing others with functions of experience so that they no longer need to be estimated. Consequently, it is econometrically simpler than the popular weighted fictitious play and reinforcement learning models. The functions of experience which replace free parameters “self-tune” over time, adjusting in a way that selects a sensible learning rule to capture subjects’ choice dynamics. For instance, the self-tuning EWA model can turn from a weighted fictitious play into an averaging reinforcement learning as subjects equilibrate and learn to ignore inferior foregone payoffs. The theory was tested on seven different games, and compared to the earlier parametric EWA model and a one-parameter stochastic equilibrium theory (QRE). Self-tuning EWA does as well as EWA in predicting behavior in new games, even though it has fewer parameters, and fits reliably better than the QRE equilibrium benchmark

It's My Turn ... Please, After You: An Experimental Study of Cooperation and Social Conventions

We introduce a class of two-player cooperation games where each player faces a binary decision, enter or exit. These games have a unique Nash equilibrium of entry. However, entry imposes a large enough negative externality on the other player such that the unique social optimum involves the player with the higher value to entry entering and the other player exiting. When the game is repeated and players' values to entry are private, cooperation admits the form of either taking turns entering or using a cutoff strategy and entering only for high private values of entry. Even with conditions that provide opportunities for unnoticed or non-punishable 'cheating', our empirical analysis including a simple strategy inference technique reveals that the Nash-equilibrium strategy is never the modal choice. In fact, most subjects employ the socially optimal symmetric cutoff strategy. These games capture the nature of cooperation in many economic and social situations such as bidding rings in auctions, competition for market share, labor supply decisions in the face of excess supply, queuing in line and courtship.cooperation, incomplete information, random payoffs, strategy inference, experimental economics.

The Evolution of Coordination under Inertia

This paper models the phenomenon of inertia driven by individual strategy switching costs in a stochastic evolutionary context. Kandori, Mailath, and Rob's (1993) model of a finite population of agents repeatedly playing a 2x2 symmetric coordination game is extended to allow for such inertia. Taking noise to the limit, a number of new short- to medium-run equilibria emerge, centred around the mixed-strategy equilibrium. Thus, unusually, an evolutionary model is seen to provide some justification for the controversial concept of mixed-strategy equilibrium. However, Kandori, Mailath, and Rob's long-run selection of the risk-dominant equilibrium continues to hold, both under fixed-rate mutations and under state-dependent mutations driven by stochastic switching costs. The key to this is the satisfaction of Blume's (1999) "skew-symmetry" of the noise process, which is shown to be crucial even under simultaneous strategy revisions. In fact, the presence of the new short-run equilibria can under certain conditions serve to reduce the expected waiting time before the risk-dominant equilibrium is reached - an instance of Ellison's (2000) idea that evolution is more rapid when it can proceed via a series of small "steps" between extremes. This suggests inertia to be a surprisingly efficient phenomenon, and also serves to moderate the force of the Ellison (1993) critique of excessively long transition times in models with vanishing noise.

Inclusive Cognitive Hierarchy

Cognitive hierarchy theory, a collection of structural models of non-equilibrium thinking, in which players' best responses rely on heterogeneous beliefs on others' strategies including naive behavior, proved powerful in explaining observations from a wide range of games. We introduce an inclusive cognitive hierarchy model, in which players do not rule out the possibility of facing opponents at their own thinking level. Our theoretical results show that inclusiveness is crucial for asymptotic properties of deviations from equilibrium behavior in expansive games. We show that the limiting behaviors are categorized in three distinct types: naive, Savage rational with inconsistent beliefs, and sophisticated. We test the model in a laboratory experiment of collective decision-making. The data suggests that inclusiveness is indispensable with regard to explanatory power of the models of hierarchical thinking.Series: Department of Strategy and Innovation Working Paper Serie