Impulse Balance Equilibrium Research Articles

Relaxing the symmetry assumption in participation games: a specification test for cluster-heterogeneity

We propose a novel approach to check whether individual behaviour in binary-choice participation games is consistent with the restrictions imposed by symmetric models. This approach allows in particular an assessment of how much cluster-heterogeneity a symmetric model can tolerate to remain consistent with its behavioural restrictions. We assess our approach with data from market-entry experiments which we analyse through the lens of ‘Exploration versus Exploration’ (EvE, which is equivalent to Logit-QRE) or of Impulse Balance Equilibrium (IBE). We find that when the symmetry assumption is imposed, both models are typically rejected when assuming pooled data and IBE yields more data-consistent estimates than EvE, i.e., IBE’s estimates of session and pooled data are more consistent than those of EvE. When relaxing symmetry, EvE (IBE) is rejected for 17% (42%) of the time. Although both models support cluster-heterogeneity, IBE is much less likely to yield over-parametrised specifications and insignificant estimates so it outperforms EvE in accommodating a model-consistent cluster-heterogeneity. The use of regularisation procedures in the estimations partially addresses EvE’s shortcomings but leaves our overall conclusions unchanged.

From Behavioral Theories to Econometrics: Inferring Preferences of Human Agents from Data on Repeated Interactions

We consider the problem of estimating preferences of human agents from data of strategic systems where the agents repeatedly interact. Recently, it was demonstrated that a new estimation method called "quantal regret" produces more accurate estimates for human agents than the classic approach that assumes that agents are rational and reach a Nash equilibrium; however, this method has not been compared to methods that take into account behavioral aspects of human play. In this paper we leverage equilibrium concepts from behavioral economics for this purpose and ask how well they perform compared to the quantal regret and Nash equilibrium methods. We develop four estimation methods based on established behavioral equilibrium models to infer the utilities of human agents from observed data of normal-form games. The equilibrium models we study are quantal-response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium, and impulse-balance equilibrium. We show that in some of these concepts the inference is achieved analytically via closed formulas, while in the others the inference is achieved only algorithmically. We use experimental data of 2x2 games to evaluate the estimation success of these behavioral equilibrium methods. The results show that the estimates they produce are more accurate than the estimates of the Nash equilibrium. The comparison with the quantal-regret method shows that the behavioral methods have better hit rates, but the quantal-regret method performs better in terms of the overall mean squared error, and we discuss the differences between the methods.

Suboptimality of Probability Matching − a Formal Proof, a Graphical Analysis and an Impulse Balance Interpretation

We prove suboptimality of probability matching in prediction tasks with an arbitrary (finite) number of outcomes and repetitions. For the popular case of binary prediction tasks, we also provide a graphical representation of the result. Finally, we relate probability matching to impulse balance equilibrium theory and show when probability matching is consistent with its predictions.

Rationalizing (Non-)equilibrium Bidding in Maximum-Value Auctions Without Beliefs About Others’ Behavior

We propose a novel approach to the modelling of second-price Maximum-Value auctions that assumes no belief about others’ behavior and no expected profit maximization. This individual decision-making model, naive Impulse Balance Equilibrium or nIBE, deals with bidders’ anticipated regrets from winning and from losing the auction. It exploits the stochastic properties of the auction format and rationalizes: (i) Nash equilibrium bidding, (ii) (non-)monotone overbidding and (iii) fully cursed equilibrium bidding. We fit this model to the available data and find that it explains median bids better than the Nash equilibrium prediction and, overall, as well as cursed-equilibrium. Furthermore, nIBE and the noise-free variant of cursed equilibrium typically outperform HQRE models with level-k or cursed equilibrium beliefs in terms of in- and out-of-sample quartile predictions.

Exploration vs Exploitation, Impulse Balance Equilibrium, and a Specification Test for the El Farol Bar Problem

The paper reports on market-entry experiments that manipulate both payoff structures and payoff levels to assess two stationary models of behaviour: Exploration vs Exploitation (EvE, which is equivalent to Quantal Response Equilibrium) and Impulse Balance Equilibrium (IBE). These models explain the data equally well in terms of goodness-of-fit whenever the observed probability of entry is less than the symmetric Nash equilibrium prediction; otherwise IBE marginally outperforms EvE. When assuming agents playing symmetric strategies, and estimating the models with session data, IBE yields more theory-consistent estimates than EvE, no matter the payoff structure or level. However, the opposite occurs when the symmetry assumption is relaxed. The conduct of a specification test rejects the validity of the restrictions on entry probabilities imposed by EvE for agents with symmetric strategies, in 50 to 75% of sessions and it always rejects it in the case of IBE, which indicates that the symmetric variant of these models has little empirical support.

A Non-Game-Theoretic Approach to Bidding in First-Price and All-Pay Auctions

We propose a novel approach to the modelling of behavior in first-price and all-pay auctions that builds on the use of a heuristic to achieve an Impulse Balance Equilibrium. The resulting individual decision-making model, nIBE, assumes no expected profit-maximization and accommodates any distribution of private values. Its parameter-free variant entails the Symmetric Bayes Nash Equilibrium bidding strategy for risk neutral bidders. Assuming impulse weighting may lead to under- or overbidding and organizes the effect of end-of-round information feedback on behavior. We assess nIBE’s explanatory power with experimental data and find that it usually outperforms the available regret models of bidding in first-price and all-pay auctions.

Exploration vs Exploitation, Impulse Balance Equilibrium, and a Specification Test for the El Farol Bar Problem

The paper reports on market-entry experiments that manipulate both payoff structures and payoff levels to assess two stationary models of behaviour: Exploration vs Exploitation (EvE, which is equivalent to Quantal Response Equilibrium) and Impulse Balance Equilibrium (IBE). These models explain the data equally well in terms of goodness-of-fit whenever the observed probability of entry is less than the symmetric Nash equilibrium prediction; otherwise IBE marginally outperforms EvE. When assuming agents playing symmetric strategies, and estimating the models with session data, IBE yields more theory-consistent estimates than EvE, no matter the payoff structure or level. However, the opposite occurs when the symmetry assumption is relaxed. The conduct of a specification test rejects the validity of the restrictions on entry probabilities imposed by EvE for agents with symmetric strategies, in 50 to 75% of sessions and it always rejects it in the case of IBE, which indicates that the symmetric variant of these models has little empirical support.

Impulse Balance and Multiple‐Period Feedback in the Newsvendor Game

Human subjects in the newsvendor game place suboptimal orders: orders are typically between the expected profit‐maximizing quantity and mean demand (“pull‐to‐center bias”). In previous work, we have shown that impulse balance equilibrium (IBE), which is based on a simple ex post rationality principle along with an equilibrium condition, can predict ordering decisions in the laboratory. In this study, we extend IBE to standing orders and multiple‐period feedback and show that it predicts—in line with previous findings—that constraining newsvendors to make a standing order for a sequence of periods moves the average of submitted orders toward the optimum.

Learning in Networks – An Experimental Study Using Stationary Concepts

Our study analyzes theories of learning for strategic interactions in networks. Participants played two of the 2 x 2 games used by Selten and Chmura (2008). Every participant played against four neighbors and could choose a different strategy against each of them. The games were played in two network structures: a lattice and a circle. We compare our results with the predictions of different theories (Nash equilibrium, quantal response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium, and impulse balance equilibrium) and the experimental results of Selten and Chmura (2008). One result is that the majority of players choose the same strategy against each neighbor. As another result we observe an order of predictive success for the stationary concepts that is different from the order shown by Selten and Chmura.

Learning in Networks—An Experimental Study Using Stationary Concepts

Our study analyzes theories of learning for strategic interactions in networks. Participants played two of the 2 × 2 games used by Selten and Chmura [1]. Every participant played against four neighbors. As a distinct aspect our experimental design allows players to choose different strategies against each different neighbor. The games were played in two network structures: a lattice and a circle. We analyze our results with respect to three aspects. We first compare our results with the predictions of five different equilibrium concepts (Nash equilibrium, quantal response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium, and impulse balance equilibrium) which represent the long-run equilibrium of a learning process. Secondly, we relate our results to four different learning models (impulse-matching learning, action-sampling learning, self-tuning EWA, and reinforcement learning) which are based on the (behavioral) round-by-round learning process. At last, we compare the data with the experimental results of Selten and Chmura [1]. One main result is that the majority of players choose the same strategy against each neighbor. As other results, we observe an order of predictive success for the equilibrium concepts that is different from the order shown by Selten and Chmura and an order of predictive success for the learning models that is only slightly different from the order shown in a recent paper by Chmura, Goerg and Selten [2].

Impulse balance in the newsvendor game

One striking behavioral phenomenon is the “pull-to-center” bias in the newsvendor game: facing stochastic demand, subjects tend to order quantities between the expected profit maximizing quantity and mean demand. We show that the impulse balance equilibrium, which is based on a simple ex-post rationality principle along with an equilibrium condition, predicts the pull-to-center bias and other, more subtle observations in the laboratory newsvendor game.

The Minority of Three-Game: An Experimental and Theoretical Analysis

We report experimental results on the minority of three-game, where three players choose one of two alternatives and the most rewarding alternative is the one chosen by a single player. This coordination game has many asymmetric equilibria in pure strategies that are non-strict and payoff-asymmetric and a unique symmetric mixed strategy equilibrium in which each player’s behavior is based on the toss of a fair coin. This straightforward behavior is predicted by equilibrium selection, impulse-balance equilibrium, and payoff-sampling equilibrium. Experimental participants rely on various decision rules, and only a quarter of them perfectly randomize.

Stationary Concepts for Experimental 2 × 2 Games: Comment

Reinhard Selten and Thorsten Chmura (2008) recently reported laboratory results for completely mixed 2 X 2 games used to compare Nash equilibrium with four other stationary concepts: quantal response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium, and impulse balance equilibrium. We reanalyze their data, correct some errors, and find that Nash clearly fits worst while the four other concepts perform about equally well. We also report new analysis of other previous experiments that illustrate the importance of the loss aversion hardwired into impulse balance equilibrium: when the other non-Nash concepts are augmented with loss aversion, they outperform impulse balance equilibrium.

An Alternative Approach to Compare Three Stationary Concepts: A Dummy Judgment

Three stationary concepts, Nash equilibrium, quantal response equilibrium and impulse balance equilibrium, have different predictions for a given completely mixed 2x2-game, which leads experimental economists to conduct relevant experiments with different parameters for testing the performances of theoretical predictions by those concepts.(see et al., 2008; Brunner et al., forthcoming and Goerg et al., forthcoming in AER etc.) Limited by the precision of the data from laboratory experiments, it is, however, usually difficult to evaluate different stationary concepts in terms of the Euclidian distance between the prediction and the data. We suggest that it's important to clarify the internal logic of those theories and determine typical discrepancies between those theories. By mathematical proof, we show that, if a Dummy is added to a constant game and thus we get a non-constant game, only the IBE in those three concepts prefigures a regular change of the predicted equilibrium, namely, if we add a diamond-shaped Dummy on the constant game, the probability of the UP used by the UP-DOWN player, compared with that in the original constant game, will increase in the new non-constant game, and, in contrast, if we add a L-shaped Dummy, the probability of UP will decrease; meanwhile, in both situations that of the LEFT used by LEFT-RIGHT player has no such regular change. However, all other theories do not logically contain such a deduction. Thus, we get another approach for testing those concepts, that is, experiments can be designed to identify whether there are stable discrepancies in those three games - the constant game and two non-constant games transformed from the constant game by adding Dummy. Accordingly, we have three treatments in this paper, the control treatment with payoff matrix of [(5, 0) (0, 5) (0, 5) (5, 0)], treatment 1 with payoff matrix of [(5, 5) (5, 10) (0, 5) (10, 0)] from adding a Diamond-shaped Dummy on the payoff matrix of the control treatment and treatment 2 with payoff matrix of [(10, 0) (0, 5) (5, 10) (5, 5)] form adding a L-shaped Dummy. For QRE, as well as other best response structure concepts, this design ensures that, for any given λ, such additive changes do not have any effect on the quantal response equilibrium. However, IBE tells a stationary state of (0.5, 0.5), (0.667, 0.5) and (0.333, 0.5) for the control treatment, treatment 1 and treatment 2 respectively, which is, in other word, that IBE tells a regular change in the frequency of UP-DOWN but no change of LEFT-RIGHT. The observed equilibria are as follows: UP frequencies are 0.511(±0.005), 0.592(±0.029) and 0.384(±0.009) for the control treatment, treatment 1 and treatment 2, respectively; LEFT frequencies are 0.492(±0.005), 0.519(±0.030) and 0.497(±0.014), respectively. Thus, we show that, in terms of predicting the regular change, IBE outperforms in those three stationary concepts. Meanwhile, further analysis of the IBE reveals that the success of IBE is due to Selten Transformation( Brunner et al. noted this point in terms of Euclidian distance.), not Impulse Balance. On this basis, the paper concludes by discussing possible ways to improve those stationary concepts.

Experimental investigation of stationary concepts in cyclic duopoly games

We experimentally test the predictive success of three stationary concepts in two cyclic duopoly games. The concepts are Nash equilibrium, impulse-balance equilibrium and payoff-sampling equilibrium. In the experiment 11 independent subject groups, consisting out of six participants interacting over 200 rounds, were gathered for each game. The comparison of the three concepts with mixed strategies shows that the order of performance from best to worst is as follows: payoff-sampling equilibrium, impulse-balance equilibrium, and Nash equilibrium. In addition the data exhibit a weak but significant tendency over time in the direction of coordination at a pure strategy equilibrium.

Incorporating Fairness Motives into the Impulse Balance Equilibrium and Quantal Response Equilibrium Concepts: An Application to 2x2 Games

Substantial evidence has accumulated in recent empirical works on the limited ability of the Nash equilibrium to rationalize observed behavior in many classes of games played by experimental subjects. This realization has led to several attempts aimed at finding tractable equilibrium concepts which perform better empirically; one such example is the impulse balance equilibrium (Selten, Chmura, 2008), which introduces a psychological reference point to which players compare the available payoff allocations. This paper is concerned with advancing two new, empirically sound, concepts: equity-driven impulse balance equilibrium (EIBE) and equity-driven quantal response equilibrium (EQRE): both introduce a distributive reference point to the corresponding established stationary concepts known as impulse balance equilibrium (IBE) and quantal response equilibrium (QRE). The explanatory power of the considered models leads to the following ranking, starting with the most successful in terms of fit to the experimental data: EQRE, IBE, EIBE, QRE and Nash equilibrium.

Stationary Concepts for Experimental 2x2-Games

Five stationary concepts for completely mixed 2x2-games are experimentally compared: Nash equilibrium, quantal response equilibrium, action-sampling equilibrium, payoff-sampling equilibrium (Martin J. Osborne and Ariel Rubinstein 1998), and impulse balance equilibrium. Experiments on 12 games, 6 constant sum games, and 6 nonconstant sum games were run with 12 independent subject groups for each constant sum game and 6 independent subject groups for each nonconstant sum game. Each independent subject group consisted of four players 1 and four players 2, interacting anonymously over 200 periods with random matching. The comparison of the five theories shows that the order of performance from best to worst is as follows: impulse balance equilibrium, payoff-sampling equilibrium, action-sampling equilibrium, quantal response equilibrium, Nash equilibrium. (JEL C70, C91)

Games of Competition in a Stochastic Environment

The paper presents a set of games of competition between two or three players in which reward is jointly determined by a stochastic biased mechanism and players’ choices. More specifically, a resource can be found with unequal probabilities in one of two locations. The first agent is rewarded only if it finds the resource and avoids being found by the next agent in line; the latter is rewarded only if it finds the former. Five benchmarks, based on different psychological and game-theoretic assumptions are derived and their predictions compared to actual behavior of 120, 40, and 48 participants playing repeatedly. Of the five benchmarks—the unique (Nash) equilibrium, reinforcement learning, trust-based efficiency, maximum unpredictability, and regret-based (Impulse Balance) equilibrium—regret for missed opportunities best accounts for the qualitative aspect of participants’ behavior and regret attenuated by randomization best accounts for the quantitative aspect of behavior.

Impulse balance equilibrium and feedback in first price auctions

Experimental sealed-bid first-price auctions with private values in which feedback on the losing bids is provided yield lower revenues than auctions where this feedback is not given. The concept of weighted impulse balance equilibrium, which is based on a principle of ex-post rationality and incorporates a concern for social comparison, captures the data.