Hierarchical Reasoning versus Iterated Reasoning in p-Beauty Contest Guessing Games

This paper analyzes strategic choice in p-beauty contests. We first show that it is not generally a best reply to guess the expected target value (accounting for the own weight) even in games with n>2 players and that iterated best response sequences are not unique even after perfect/cautious refinement. This implies that standard formulations of ``level-k'' models are neither exactly nor uniquely rationalizable by belief systems based on iterated best response. Second, exact modeling of iterated reasoning weakens the fit considerably and reveals that equilibrium types dominate the populations. We also show that ``levels of reasoning'' cannot be measured regardless of the underlying model. Third, we consider a ``nested logit'' model where players choose their level. It dispenses with belief systems between players and is rationalized by a random utility model. Besides being internally consistent, nested logit equilibrium fits better than three variants of the level-k model in standard data sets.logit equilibrium, hierarchical response, level-k, beauty contest

Iterated dominance and iterated best response in experimental "p-beauty contests"

Picture a thin country 1000 miles long, running north and south, like Chile. Several natural attractions are located at the northern tip of the country. Suppose each of n resort developers plans to locate a resort somewhere on the country's coast (and all spots are equally attractive). After all the resort locations are chosen, an airport will be built to serve tourists, at the average of all the locations including the natural attractions. Suppose most tourists visit all the resorts equally often, except for lazy tourists who visit only the resort closest to the airport; so the developer who locates closest to the airport gets a fixed bonus of extra visitors. Where should the developer locate to be nearest to the airport? The surprising game-theoretic answer is that all the developers should locate exactly where the natural attractions are. This answer requires at least one natural attraction at the northern tip, but does not depend on the fraction of lazy tourists or the number of developers (as long as there is more than one)

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

Beauty Contested. How much of Keynes’ remains in Behavioural Economics Beauty Contests?

In one of the most famous passages of economic literature, John Maynard Keynes (1936, p.156) likens the stock market to a beauty contest in which the winners are those who anticipate the average opinion. Recently there have been attempts at investigating the BC experimentally (Nagel 1995, Duffy & Nagel 1997, Ho et al. 1998, Bosch-Domenech et al. 2002, Güth et al. 2002). In Experimental Beauty Contests, participants choose a real number from a closed interval, e.g. I [0,100]. Whoever picks the number closest to p times the average (usuallywith p = 2/3) is the winner of a monetary reward. An experiment like this is dominance solvable: the process of iterated elimination of dominated strategies leads to the unique and stable equilibrium at which every player chooses zero, and every player wins. Keynes’ metaphor, on the other hand, referred to a situation in which not all participants can win, so that the goal of individual investors and speculators must be “to outwit the crowd” (p. 152). Despite the differences, the Keynesian theory of decision under uncertainty tallies with the behaviour observed in Experimental Beauty Contests.Beauty Contest, Behavioural Economics, Keynes, Reasoning.

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

Cognitive effort in the Beauty Contest Game

This paper analyzes cognitive effort in 6 different one-shot p-beauty games. We use both Raven and Cognitive Reflection tests to identify subjects' abilities. We find that the Raven test does not provide any insight on beauty contest game playing but CRT does: subjects with higher scores on this test are more prone to play dominant strategies.Beauty Contest Game, Raven, Cognitive Reflection Test

Billiards and Brains: Cognitive Ability and Behavior in a p-Beauty Contest

"Beauty contests" are well-studied, dominance-solvable games that generate two interesting results. First, most behavior does not conform to the unique Nash equilibrium. Second, there is considerable unexplained heterogeneity in behavior. In this work, we evaluate the relationship between beauty contest behavior and cognitive ability. We find that subjects with high cognitive ability exhibit behavior that is closer to the Nash equlibrium. We examine this finding through the prism of economic and biological theory.beauty contest; rationality; cognitive ability; Nash equlibrium

Progress in Behavioral Game Theory

Is game theory meant to describe actual choices by people and institutions or not? It is remarkable how much game theory has been done while largely ignoring this question. The seminal book by von Neumann and Morgenstern, The Theory of Games and Economic Behavior, was clearly about how rational players would play against others they knew were rational. In more recent work, game theorists are not always explicit about what they aim to describe or advise. At one extreme, highly mathematical analyses have proposed rationality requirements that people and firms are probably not smart enough to satisfy in everyday decisions. At the other extreme, adaptive and evolutionary approaches use very simple models-mostly developed to describe nonhuman animals-in which players may not realize they are playing a game at all. When game theory does aim to describe behavior, it often proceeds with a disturbingly low ratio of careful observation to theorizing