A cognitive hierarchy model of games

Players in a game are “in equilibrium” if they are rational, and accurately predict other players' strategies. In many experiments, however, players are not in equilibrium. An alternative is “cognitive hierarchy” (CH) theory, where each player assumes that his strategy is the most sophisticated. The CH model has inductively defined strategic categories: step 0 players randomize; and step k thinkers best-respond, assuming that other players are distributed over step 0 through step k − 1. This model fits empirical data, and explains why equilibrium theory predicts behavior well in some games and poorly in others. An average of 1.5 steps fits data from many games

A cognitive hierarchy theory of one-shot games: Some preliminary results

Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modelling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in in one-shot games before they can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games. The CH approach assumes that players use k steps of reasoning with frequency f (k). Zero-step players randomize. Players using k (≥ 1) steps best respond given partially rational expectations about what players doing 0 through k - 1 steps actually choose. A simple axiom which expresses the intuition that steps of thinking are increasingly constrained by working memory, implies that f (k) has a Poisson distribution (characterized by a mean number of thinking steps τ ). The CH model converges to dominance-solvable equilibria when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts effects of group size which are not predicted by Nash equilibrium. Best-fitting values of τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable p-beauty contests. The CH model also has economic value because subjects would have raised their earnings substantially if they had best-responded to model forecasts instead of making the choices they did

No Magic for Market Entry in the Field: Evidence from Taxi Markets

We study taxi markets in Singapore to understand market entry in the field. While market entry games in the lab consistently produce equilibrium outcomes, we show that a lack of market knowledge hinders the markets from consistently reaching equilibrium in the field. In Singapore, a small 720-square-kilometre island city that can be divided into 29 taxi markets, full equilibrium is elusive: 68% of the market-entry decisions made by the 2,728 taxi drivers in our data could be improved. Using three months of earnings and detailed movement data from these taxi drivers, we find an average 20\% gap in marginal wage rates across markets. We use dynamic programming to derive the optimal solution for more than 3 million search decisions and find that only 32% of the searches ended in an optimal market. Finally, we find that market knowledge developed in a given month explains an additional 3% variation of the earning losses in the 2.6 million decisions for the subsequent two months, an improvement in model fit of 74%; while strategic thinking and minimization risk have no impact on earning loss

Behavioral Game Theory: Thinking, Learning and Teaching

Game theory is a mathematical system for analysing and predicting how humans behave in strategic situations. Standard equilibrium analyses assume that all players: (1) form beliefs based on an analysis of what others might do (strategic thinking); (2) choose the best response given those beliefs (optimization); and (3) adjust best responses and beliefs until they are mutually consistent (equilibrium)

Yellow taxis have fewer accidents than blue taxis because yellow is more visible than blue

Is there a link between the color of a taxi and how many accidents it has? An analysis of 36 mo of detailed taxi, driver, and accident data (comprising millions of data points) from the largest taxi company in Singapore suggests that there is an explicit link. Yellow taxis had 6.1 fewer accidents per 1,000 taxis per month than blue taxis, a 9% reduction in accident probability. We rule out driver difference as an explanatory variable and empirically show that because yellow taxis are more noticeable than blue taxis—especially when in front of another vehicle, and in street lighting—other drivers can better avoid hitting them, directly reducing the accident rate. This finding can play a significant role when choosing colors for public transportation and may save lives as well as millions of dollars

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

A parsimonious model of stockkeeping-unit choice

10.1509/jmkr.40.3.351.19232Journal of Marketing Research403351-36