The Theory of Theft: An Inspection Game Model of the Stolen Base Play in Baseball

This paper applies the theory of equilibrium in mixed strategies in an inspection game model to describe the strategic interaction in the stolen base play in baseball. A parsimonious simultaneous-move game model offers predictions about how the observable conduct of the teams on offense and defense responds as the characteristics of the players involved change. The theory organizes observations from play-by-play data from Major League Baseball, where highly-motivated, experienced professionals interact in an environment where private information is not significant.mixed strategy, Markov equilibrium, baseball

A Dynamic Homotopy Interpretation of Quantal Response Equilibrium Correspondences

This paper uses properties of the logistic quantal response equilibrium correspondence to compute Nash equilibria in nite games. It is shown that branches of the correspondence may be numerically traversed e ciently and securely. The method can be implemented on a multicomputer, allowing for application to large games. The path followed by the method has an interpretation analogous to Harsanyi and Selten's Tracing Procedure. As an application, it is shown that the principal branch of any quantal response equilibrium correspondence satisfying a monotonicity property converges to the risk-dominant equilibrium in 2x2 games.noncooperative games, computation of Nash equilibrium, quantal response, logit equilibrium.

Mathematics self-confidence and the "prepayment effect" in riskless choices

We extend the analysis of a riskless choice experiment reported recently by Hochman et al. (2014). Participants select from among sets of standard playing cards valued by a simple formula. In some sessions, participants are given a prepayment associated with some of the cards, which need not be the earnings-maximizing ones. Hochman et al. find that participants choose an earnings-maximizing card less frequently when another card is prepaid. We replicate this result under the original instructions, but not with instructions which explain the payment process more explicitly. Participants who state they do not consider themselves good at mathematics make earnings-maximizing choices much less frequently overall, but those who express self-confidence in mathematics drive the treatment effect. The results suggest that even when comparisons among choices require only simple quantitative reasoning steps, market designers and regulators may need to pay close attention to how the terms of offers are expressed, explained, and implemented

Is Batting Last an Advantage?

This paper applies the theory of zero-sum stochastic games to assess the validity of baseball's ancient wisdom that batting last confers a strategic advantage. Results from numerical calculation of Markov perfect equilibrium suggest that the team that bats last will have an advantage if in fact the offense has, in some sense, more useful strategic actions available than the defense. An example is provided where the advantage depends on details of the teams playing. Regardless of which team has the advantage, all calculations indicate the advantage is negligible in magnitude.zero-sum games, Markov perfect equilibrium, baseball

Offensive Performance, Omitted Variables, and the Value of Speed in Baseball

This note considers the problem of estimating the marginal products of offensive events towards a baseball team's objective of scoring runs. Regression techniques on official statistics give a positive marginal product for a stolen base attempt, which is inconsistent with the theory of mixed strategy Nash equilibrium. Augmenting the specification of the production function to include other productive qualities of footspeed restores estimates consistent with equilibrium theory.Omitted variable bias, mixed strategies, equilibrium test, baseball

Correlation neglect and case-based decisions

In most theories of choice under uncertainty, decision-makers are assumed to evaluate acts in terms of subjective values attributed to consequences and probabilities assigned to events. Case-based decision theory (CBDT), proposed by Gilboa and Schmeidler, is fundamentally different, and in the tradition of reinforcement learning models. It has no state space and no concept of probability. An agent evaluates each available act in terms of the consequences he has experienced through choosing that act in previous decision problems that he perceives to be similar to his current problem. Gilboa and Schmeidler present CBDT as a complement to expected utility theory (EUT), applicable only when the state space is unknown. Accordingly, most experimental tests of CBDT have used problems for which EUT makes no predictions. In contrast, we test the conjecture that case-based reasoning may also be used when relevant probabilities can be derived by Bayesian inference from observations of random processes, and that such reasoning may induce violations of EUT. Our experiment elicits participants’ valuations of a lottery after observing realisations of the lottery being valued and realisations of another lottery. Depending on the treatment, participants know that the payoffs from the two lotteries are independent, positively correlated, or negatively correlated. We find no evidence of correlation neglect indicative of case-based reasoning. However, in the negative correla- tion treatment, valuations cannot be explained by Bayesian reasoning, while stated qualitative judgements about chances of winning can

Wage bargaining with direct competition and heterogeneous access to vacancies

Abstract Agents with a richer set of opportunities to trade should be able to demand better terms of trade. For instance, workers who are otherwise equally-qualified may differ in their access to vacancies, e.g. because their social networks are larger or smaller. We present a model of search and matching in which multiple workers may be matched to the same vacancy, and workers compete directly in the wage bargining process. Workers with greater access have a higher dynamic outside option and demand higher wages. They are therefore unsuccessful candidates in some matches; this latter outcome is not possible in existing models based on Nash bargaining to determine wages. In particular, when markets are tight and the expected length of a position is short, workers with better access to opportunities will remain unemployed longer than those with less access

Majoritarian Contests with Asymmetric Battlefields: An Experiment

We investigate a version of the classic Colonel Blotto game in which individual battles may have different values. Two players allocate a fixed budget across battlefields and each battlefield is won by the player who allocates the most to that battlefield. The winner of the game is the player who wins the battlefields with highest total value. We focus on the case where there is one large and several small battlefields, such that a player wins if he wins the large and any one small battlefield, or all the small battlefields. We compute the mixed strategy equilibrium for these games and compare this with choices from a laboratory experiment. The equilibrium predicts that the large battlefield receives more than a proportional share of the resources of the players, and that most of the time resources should be spread over more battlefields than are needed to win the game. We find support for the main qualitative features of the equilibrium. In particular, strategies that spread resources widely are played frequently, and the large battlefield receives more than a proportional share in the treatment where the asymmetry between battlefields is stronger.We thank Subhasish Chowdhury, Judith Avrahami, seminar participants at New York University, Keele University, University of East Anglia, University of the Basque Country, and conference participants at the Voting Power in Practice Symposium at LSE 2011, M-BEES 2011, SING7 2011, Contest, Mechanisms and Experiments Conference at Exeter 2012, SAET 2012, GAMES 2012 and ESEM-EEA 2013. The equilibrium computations were carried out on the High Performance Computing Cluster supported by the Research and Specialist Computing Support service at the University of East Anglia

Two-bidder all-pay auctions with interdependent valuations, including the highly competitive case

We analyze symmetric, two-bidder all-pay auctions with interdependent valuations and discrete type spaces. Relaxing previous restrictions on the distribution of types and the valuation structure, we present a construction that characterizes all symmetric equilibria. We show how the search problem this construction faces can be complex. In equilibrium, randomization can take place over disjoint intervals of bids, equilibrium supports can have a rich structure, and non-monotonicity of the equilibrium may result in a positive probability of allocative inefficiency when the value of the prize is not common. Particular attention is paid to the case in which an increase in a bidder’s posterior expected value of winning the auction is likely to be accompanied by a corresponding increase for the other bidder. Such environments are “highly competitive” in the sense that the bidder’s higher valuation also signals that the other bidder has an incentive to bid aggressively

Majoritarian Blotto contests with asymmetric battlefields: an experiment on apex games

We investigate a version of the classic Colonel Blotto game in which individual battlefields may have different values. Two players allocate a fixed discrete budget across battlefields. Each battlefield is won by the player who allocates the most to that battlefield. The player who wins the battlefields with highest total value receives a constant winner payoff, while the other player receives a constant loser payoff. We focus on apex games, in which there is one large and several small battlefields. A player wins if he wins the large and any one small battlefield, or all the small battlefields. For each of the games we study, we compute an equilibrium and we show that certain properties of equilibrium play are the same in any equilibrium. In particular, the expected share of the budget allocated to the large battlefield exceeds its value relative to the total value of all battlefields, and with a high probability (exceeding 90% in our treatments) resources are spread over more battlefields than are needed to win the game. In a laboratory experiment, we find that strategies that spread resources widely are played frequently, consistent with equilibrium predictions. In the treatment where the asymmetry between battlefields is strongest, we also find that the large battlefield receives on average more than a proportional share of resources. In a control treatment, all battlefields have the same value and our findings are consistent with previous experimental findings on Colonel Blotto games