Quantal Response Equilibrium Research Articles

Non-parametric identification and testing of quantal response equilibrium

This paper studies the falsifiability and identification of Quantal Response Equilibrium (QRE) when each player's utility and error distribution are relaxed to be unknown non-parametric functions. Using the variation of players' choices across a series of games, we first show that both the utility function and the distribution of errors are non-parametrically over-identified. This over-identification result further suggests a straightforward testing procedure for QRE which achieves the desired type-1 error and maintains a small type-2 error. To apply this methodology, we conduct an experimental study of the matching pennies game. Our non-parametric estimates strongly reject the conventional Logit choice probability. Moreover, when the utility and the error distribution are sufficiently flexible and heterogeneous, the quantal response hypothesis cannot be rejected for 70% of participants. However, strong assumptions such as linear utility, logistically distributed errors, and homogeneity lead to substantially higher rejection rates.

A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium

A Variant of the Logistic Quantal Response Equilibrium to Select a Perfect Equilibrium

Quantal Response Equilibrium and Rationalizability: Inside the Black Box

This paper aims to connect epistemic and behavioral game theory by examining the epistemic foundations of quantal response equilibrium (QRE) in static games. We focus on how much information agents possess about the probability distributions of idiosyncratic payoff shocks, in addition to the standard assumptions of rationality and common belief in rationality. When these distributions are transparent, we obtain a solution concept called Δp-rationalizability, which includes action distributions derived from QRE; we also give a condition under which this relationship holds true in reverse. When agents only have common belief in the monotonicity of these distributions (for example, extreme value distributions), we obtain another solution concept called ΔM-rationalizability, which includes action distributions derived from rank-dependent choice equilibrium, a parameter-free variant of QRE. Our solution concepts also provide insights for interpreting experimental and empirical data.

Pork barrel politics, voter turnout, and inequality: An experimental study

We experimentally study pork barrel politics in repeated two-candidate majoritarian elections with costly voting. Candidates form distinct supporter groups by favoring some voters in budget spending and ignoring others. Relative to compulsory voting, voluntary voting induces on average larger, more narrowly targeted favors and therefore more inequality among otherwise identical voters. Against our prediction, candidate participants tend to cultivate policy polarization by repeatedly favoring their exclusive supporters and avoiding those of the opponent. With compulsory voting, they tolerate some additional policy overlap for a separate subset of voters. Relating to theory, actual levels and patterns of voluntary turnout are well captured by quantal response equilibrium (QRE) for a noise parameter estimate external to our experiment. Importantly, and surprisingly, for reasonable assumption we find unique QRE solutions for our polity games, which include the policymaking stage and each of the many possible elections that can emerge from that stage.

Approximate computation and estimation of quantal response equilibrium through simulation

Approximate computation and estimation of quantal response equilibrium through simulation

확장된 블로토 대령 게임에서 최적 전략의 수학적 분석과 시각화

The Colonel Blotto Game is a game theory that can be applied to various social phenomena in reality, and prior research on various variations is also being conducted. This study attempted to build a strategy selection model using the concept of Quantal Response Equilibrium, an extension of the Nash Equilibrium, in the Colonel Blotto Game that assigned weights to each battlefield arrangement, identify the effectiveness of different strategies, and visualize the results. For this purpose, the rules and progression of the Colonel Blotto Game were mathematically defined and the optimal probabilistic strategy was analyzed. The results obtained using C++ were visualized and presented in two ways (a method based on troop deployment characteristic indicators and a method applying the PCA algorithm for dimensionality reduction). It was found that the smaller the model’s strategy selection volatility, the more dominant the equal distribution strategy is, and as the volatility increases, several characteristic strategies, including the equal distribution strategy, form a cyclical relationship.

Stackelberg Security Game for Optimizing Cybersecurity Decisions in Cloud Computing

As it is difficult to cover all cybersecurity threats, an optimal defense strategy is one of the focal issues in cloud computing due to its dynamic abstraction and scalability. On this basis, Stackelberg security games (SSG) have received significant attention for their better deployment of limited security. To deal with uncertainty and incomplete information, we introduce a modified quantal response (Mod-QR) approach that incorporates bounded rationality and preference into the decision-making process. Formally, this can be done by using the quantal response equilibrium (QRE) framework to find a trade-off between the effectiveness and operating costs of cloud computing. In this case, the most effective countermeasures to defend the cloud can be viewed as a mixed strategy in which all the actions of the defender are played with a nonzero probability. This framework has been evaluated using an experimental study on MATLAB optimization toolbox to understand the behavioral aspects of cybersecurity actors and then to proactively protect cloud computing.

Scoring rules in experimental procurement

We report the results of an experiment where subjects compete for procurement contracts to be awarded by means of a scoring auction. Two experimental conditions are considered, depending on the relative weight of quality vs price in the scoring rule. We show that different quality-price weights dramatically alter the strategic environment and affect efficiency. Our evidence shows that each weighting better delivers against a matching objective function than using a scoring rule which misrepresents the buyer's objective function. Nonetheless, there are large deviations in how each performs, with the higher weight on quality delivering much greater efficiency evaluated against its own objective function than a low weight on quality evaluated against its own objective function, despite the higher quality weight inducing higher deviations from equilibrium. We propose a “mediation analysis” to show that the “direct effect” (due to the different strategic properties of the induced game-forms) outweighs the “indirect” one (how the different game-forms affect out-of-equilibrium behaviour). We also perform a structural estimation of the Quantal Response Equilibrium induced by subjects’ behavior, where we find that subjects are risk averse and noisy play affects behavior in the direction of underbidding.

Rent dissipation and streamlined costs: Laboratory experiments

The effect of group size and effort cost on rent dissipation in the standard Tullock rent seeking model is examined using a laboratory experiment. Data support qualitative comparative static Nash equilibrium predictions as reductions in both group size and effort cost increase effort levels. However, effort and dissipation rates are above Nash predictions for all treatments. In addition, while cutting per-unit effort costs in half increases individual effort levels, they do not fully double as predicted and therefore lower rent dissipation, creating a policy-relevant “streamlining effect”. A quantal response equilibrium, which adds stochastic elements into the game structure, predicts the observed higher effort levels and dissipation but is proven to be unable to explain the streamlining effect. A modification of quantal response that incorporates implementation error noise and loss aversion outperforms other estimated models and tracks the observed all data patterns more closely, including the streamlining effect.

Asymmetric volunteer's dilemma game: Theory and experiment

This study explores asymmetric volunteer dilemma games in which players incur different costs for volunteering. Diekmann (1993) conjectures that a player with less cost is more likely to contribute at the equilibrium if it is risk dominant. We have re-examined this hypothesis theoretically and experimentally and found that even when focusing on risk-dominant equilibria, the explanatory power of the theory is not universal. An econometric comparison among several behavioral models including regret aversion shows that quantal response equilibrium best explains the data.

Revisiting the Asymmetric Matching Pennies Contradiction in China.

The asymmetric matching pennies contradiction posits that contrary to the prediction of mixed-strategy Nash equilibrium, experimental subjects' choices are, in practice, based heavily on the magnitudes of their own payoffs. Own-payoff effects are robustly confirmed in the literature. Closely following the experimental setups in the literature which support the contradiction, we conduct a series of asymmetric matching pennies games in China, hypothesizing play which is closer to equilibrium frequencies than previously found. Contrary to previous experiments which were conducted in the United States, we find that there are essentially no own-payoff effects among Row players who face large payoff asymmetry. In a Quantal Response Equilibrium framework allowing for altruism or spite, the behavior of our subjects corresponded to a positive spite parameter, whereas the results of previous studies corresponded to altruism. Our results may be consistent with recent psychology literature that finds people from collectivist cultures are substantially more adept at taking the perspective of others compared with people from individualist cultures, a feature of the reasoning needed to obtain mixed-strategy equilibrium.

Abstention and informedness in nonpartisan elections

We develop and experimentally test a model of voter information acquisition in nonpartisan elections, both with and without abstention. We theoretically demonstrate that allowing for abstention can increase information acquisition, provided the cost of information is not too low. Our experimental data find that voters are less responsive than predicted to increases in the cost of information. As a result, the cost required to yield higher levels of informedness when abstention is allowed is higher than predicted. Our data are well explained by agent quantal response equilibrium, which accounts for the fact that uninformed voters are often observed to vote, even when it is not rational to do so.

Bilateral Conflict: An Experimental Study of Strategic Effectiveness and Equilibrium

AbstractBilateral conflict involves an attacker with several alternative attack methods and a defender who can take various actions to better respond to different types of attack. These situations have wide applicability to political, legal, and economic disputes, but they are particularly challenging to study empirically because the payoffs are unknown. Moreover, each party has an incentive to behave unpredictably, so theoretical predictions are stochastic. This article reports results of an experiment where the details of the environment are tightly controlled. The results sharply contradict the Nash equilibrium predictions about how the two parties’ choice frequencies change in response to the relative effectiveness of alternative attack strategies. In contrast, nonparametric quantal response equilibrium predictions match the observed treatment effects. Estimation of the experimentally controlled payoff parameters across treatments accurately recovers the true values of those parameters with the logit quantal response equilibrium model but not with the Nash equilibrium model.

Algorithms for a risk-averse Stackelberg game with multiple adversaries

We consider a Stackelberg game that arises in a security domain (SSG), where a defender can simultaneously protect m out of n targets from an adversary that observes the defense strategy before deciding on an utility maximizing attack. Given the high stakes in security settings, it is reasonable that the defender in this game is risk averse with respect to the attacker’s decisions.Here we focus on developing efficient solution algorithms for a specific SSG, where the defender uses an entropic risk measure to model risk aversion to the attacker’s strategies, and where multiple attackers select targets following logit quantal response equilibrium models. This problem can be formulated as a nonconvex nonlinear optimization problem. We propose two solution methods: (1) approximate the problem through convex mixed integer nonlinear programs (MINR) and (2) a general purpose methodology (CELL) to optimize nonconvex and nonseparable fractional problems through mixed integer linear programming approximations. Both methods provide arbitrarily good incumbents and lower bounds on SSG. We present cutting plane methods to solve these problems for large instances. Our computational experiments illustrate the advantages of introducing risk aversion into the defender’s behavior and show that MINR dominates CELL, producing in 2 h solutions that are within 2% of optimal on average.

On the smooth unfolding of bifurcations in quantal-response equilibria

We report on the topological structure of logit quantal response equilibria for asymmetric, two-player, two-choice games in normal form, via catastrophe theory. We present three main outcomes: first, a novel, smooth potential function for the underlying dynamics, relative to which logit equilibria arise as stationary points comprising a parameterised hyper-surface; secondly, a proof that all catastrophes within this manifold are of order at most three, i.e. “folds and cusps”; thirdly, discovery of a new topological phenomenon, the “pleated loop”, corresponding to a pair of opposed pitchfork bifurcations. This we exhibit for games, such as Prisoner's Dilemma, that have a unique Nash equilibrium, with multiple logit equilibria for finite precision parameters. These results extend work at the intersection of stochastic decision theory and catastrophe theory with economic games. Their application to the problem of bifurcations in the tracing procedure for logit solutions, proposed by McKelvey and Palfrey, is discussed throughout.

Decentralized Inventory Transshipments with Quantal Response Equilibrium

Despite the benefits of inventory transshipment, numerous behavioral experiments have revealed that retailers often deviate from the Nash-equilibrium ordering quantities, which in turn impacts the potential advantages. Motivated by this issue, we developed a behavioral model to analyze the deviation of ordering quantities among two independent retailers who engage in inventory transshipment from the perspective of analytical modeling. In our model, we incorporated bounded rationality with the quantal response equilibrium. Firstly, we established the existence of such a quantal response equilibrium and provided the conditions for its uniqueness. Secondly, we compared the quantal response equilibrium with the Nash equilibrium within a certain range of transshipment prices and observed that the limiting quantal response equilibrium is equivalent to the Nash equilibrium. Lastly, we design an iterative algorithm that incorporates the learning effects of the retailers to determine the quantal response equilibrium for the ordering quantity. The results indicate that the optimal ordering quantity and the nearby ordering quantities should be chosen with higher probabilities. Additionally, the retailer should gradually enhance their cognitive or computational abilities through repeated transshipment games to improve their decision-making process. Furthermore, to ensure a balanced inventory-sharing system, the evaluation of inventory strategies should consistently prioritize avoiding surplus instead of shortage.

Constrained school choice: an experimental QRE analysis

The theoretical literature on public school choice proposes centralized mechanisms that assign children to schools on the basis of parents’ preferences and the priorities children have for different schools. The related experimental literature analyzes in detail how various mechanisms fare in terms of welfare and stability of the resulting matchings, yet often provides only aggregate statistics of the individual behavior that leads to these outcomes (i.e., the degree to which subjects tell the truth in the induced simultaneous move game). In this paper, we show that the quantal response equilibrium (QRE) adequately describes individual behavior and the resulting matching in three constrained problems for which the immediate acceptance mechanism and the student-optimal stable mechanism coincide. Specifically, the comparative statics of the logit-QRE with risk-neutral and expected-payoff-maximizing agents capture the directional changes of subject behavior and the prevalence of the different stable matchings when cardinal payoffs (i.e., relative preference intensities) are modified in the experiment.

Bounded rationality for relaxing best response and mutual consistency: the quantal hierarchy model of decision making

While game theory has been transformative for decision making, the assumptions made can be overly restrictive in certain instances. In this work, we investigate some of the underlying assumptions of rationality, such as mutual consistency and best response, and consider ways to relax these assumptions using concepts from level-k reasoning and quantal response equilibrium (QRE) respectively. Specifically, we propose an information-theoretic two-parameter model called the quantal hierarchy model, which can relax both mutual consistency and best response while still approximating level-k, QRE, or typical Nash equilibrium behavior in the limiting cases. The model is based on a recursive form of the variational free energy principle, representing higher-order reasoning as (pseudo) sequential decision-making in extensive-form game tree. This representation enables us to treat simultaneous games in a similar manner to sequential games, where reasoning resources deplete throughout the game-tree. Bounds in player processing abilities are captured as information costs, where future branches of reasoning are discounted, implying a hierarchy of players where lower-level players have fewer processing resources. We demonstrate the effectiveness of the quantal hierarchy model in several canonical economic games, both simultaneous and sequential, using out-of-sample modelling.

Bayesian inference for Quantal Response Equilibrium in normal-form games

This paper develops a framework for estimating Quantal Response Equilibrium models from experimental data using Bayesian techniques. Bayesian techniques offer some advantages over the more commonly-used maximum likelihood approach: (i) more favorable small-sample properties, and (ii) ease of handling unobservable heterogeneity. As Quantal Response Equilibrium is a non-linear model, I also discuss some issues with choosing appropriate priors.Of the twenty-nine models taken to the data, the selected model assumes between-game heterogeneity in choice precision parameter λ, and some dispersion around the Quantal Response Equilibrium. Some implications of this model are discussed.

Joint location and pricing optimization of self-service in urban logistics considering customers’ choice behavior

Self-services by widely-deployed stations allow customers to pick up their packages themselves and have great potential to reduce last-mile delivery cost by aggregating spatial demands. An efficient self-service design for the locations of self-service stations and the service price will help to shift customers from the home-delivery service to the cost-effective self-service, thus reducing the vehicle routing cost of the dedicated fleet. This study proposes a comprehensive modeling and optimization framework for the joint self-service station location and service price (SLSP) optimization problem considering customers’ choice behavior and vehicle routing in order to maximize the profit of the logistics service providers in the last-mile delivery system. Logit equilibrium and Wardrop equilibrium conditions are employed to characterize the delivery service choice and route choice behavior of customers. A convex optimization model is developed to capture the distribution of customers opting for home-delivery service and self-services at various locations, who aim to maximize their expected utilities. We then formulate the SLSP problem as a mathematical program with complementarity constraints. An active set algorithm is proposed to solve the model effectively. Numerical experiments based on the Nguyen-Dupuis and Sioux Falls networks are conducted to demonstrate the proposed framework and derive managerial insights.