Standard Solution Concepts Research Articles

Incomplete cooperative games with player-centered information

The computation of a solution concept of a cooperative game usually employs values of all coalitions. However, in some applications, the values of some of the coalitions might be unknown due to high costs associated with their determination or simply because it is not possible to determine them exactly. We introduce a method to approximate standard solution concepts based only on partial characteristic function of the cooperative game. In this paper, we build on our previous results and generalise the results of our methods to a significantly larger class of structures of incomplete information.

Assortative Matching with Externalities and Farsighted Agents

We consider a one-to-one assortative matching problem in which matched pairs compete for a prize. With externalities, the standard solution concept—pairwise stable matching—may not exist. In this paper, we consider farsighted agents and analyze the largest consistent set (LCS) of Chwe (J Econ Theory 63:299-325, 1994). Despite the assortative structure of the problem, LCS tend to be large with the standard effectiveness functions: LCS can be the set of all matchings, including an empty matching with no matched pair. By modifying the effectiveness function motivated by Knuth (Marriages stables. Les Presses de l’Universite de Montreal, Montreal, 1976), LCS becomes a singleton of the positive assortative matching. Our results suggest that the choice of the effectiveness function can significantly impact the solution for a matching problem with externalities.

Exploration-exploitation in multi-agent learning: Catastrophe theory meets game theory

Exploration-exploitation is a powerful and practical tool in multi-agent learning (MAL); however, its effects are far from understood. To make progress in this direction, we study a smooth analogue of Q-learning. We start by showing that our learning model has strong theoretical justification as an optimal model for studying exploration-exploitation. Specifically, we prove (1) that smooth Q-learning has bounded regret in arbitrary games for a cost model that explicitly balances game-rewards and exploration-costs, i.e., costs from testing potentially suboptimal actions, and (2) that it always converges to the set of quantal-response equilibria (QRE), the standard solution concept for games with bounded rationality, in arbitrary weighted potential games. In our main task, we then turn to measure the effect of exploration on collective system performance. We characterize the geometry of the QRE surface in low-dimensional MAL systems and link our findings with catastrophe (bifurcation) theory. In particular, as the exploration hyperparameter evolves over-time, the system undergoes phase transitions where the number and stability of equilibria can change radically given an infinitesimal change to the exploration parameter. Based on this, we provide a formal theoretical treatment of how tuning the exploration parameter can provably lead to equilibrium selection with both positive as well as negative (and potentially unbounded) effects to system performance.

The role of the non-productive players in cooperative games with transferable utilities: a survey

Decision support systems apply the models of cooperative games with transferable utilities (TU games) for aggregating and ranking attributes based on their interdependent features. In TU games, players work together for a common goal under binding agreements. The contributions by the players to achieve this common goal generate some worth. A solution is a distribution of the total worth generated by all the players together. The Shapley value is the most popular solution of TU games. When cooperative game theoretic models are used in decision support systems, especially in Artificial Intelligence and Machine Learning models, the Shapley value is frequently used as a suitable solution, partly because it is easy to understand and partly because of its marginalistic approach. However, alternative criteria also exist in the literature that can be of interest to the practitioners. Most of the solutions based on these criteria differ by the property of a “non-productive player” whose contribution is marginal in different contexts of the game. We study how this property is used to characterize standard solution concepts of TU games. This, we believe will motivate the decision maker to explore possible alternative solutions that would treat the non-productive players differently from the Shapley assumptions.

Exploration-Exploitation in Multi-Agent Learning: Catastrophe Theory Meets Game Theory

Exploration-exploitation is a powerful and practical tool in multi-agent learning (MAL), however, its effects are far from understood. To make progress in this direction, we study a smooth analogue of Q-learning. We start by showing that our learning model has strong theoretical justification as an optimal model for studying exploration-exploitation. Specifically, we prove that smooth Q-learning has bounded regret in arbitrary games for a cost model that explicitly captures the balance between game and exploration costs and that it always converges to the set of quantal-response equilibria (QRE), the standard solution concept for games under bounded rationality, in weighted potential games with heterogeneous learning agents. In our main task, we then turn to measure the effect of exploration in collective system performance. We characterize the geometry of the QRE surface in low-dimensional MAL systems and link our findings with catastrophe (bifurcation) theory. In particular, as the exploration hyperparameter evolves over-time, the system undergoes phase transitions where the number and stability of equilibria can change radically given an infinitesimal change to the exploration parameter. Based on this, we provide a formal theoretical treatment of how tuning the exploration parameter can provably lead to equilibrium selection with both positive as well as negative (and potentially unbounded) effects to system performance.

Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics

The Uniqueness of a Mean Field Equilibrium

Interactive Unawareness in the Graph Model for Conflict Resolution

In this paper, we modify the graph model for conflict resolution (GMCR) to model interactive unawareness of decision makers (DMs) about the options available to them in the conflict. More specifically, we consider a GMCR with two DMs, where a DM, in some given state, can be unconscious about some of his options, or about the options of his opponent, and therefore, may have only a partial knowledge of the state space. By interactive unawareness, it is meant that a DM can reason about the awareness level of the other DM and about the awareness level of the other DM regarding his or her own awareness level and so on. We generalize standard solution concepts for this model and illustrate its usefulness by means of a hypothetical war conflict.

The graph model for conflict resolution with consensus fuzzy preferences and its application to environmental sustainable development

Conflict of environmental sustainable development as a common phenomenon can be seen everywhere in life. To capture consensus problems of decision makers (DMs) in conflict, a consensus and non-consensus fuzzy preference relation (FPR) matrix is proposed to the framework of the Graph Model for Conflict Resolution (GMCR). Concentrating on the case of two DMs within GMCR paradigm, four standard fuzzy solution concepts are developed into eight fuzzy stability definitions which can fully represent DMs’ behavior characteristics of win-win and self-interested. To demonstrate how the novel GMCR methodology proposed in this paper can be conveniently utilized in practice, it is then applied to an environmental sustainable development conflict with two DMs. The results show that the general fuzzy equilibrium solutions are the intersection of consensus fuzzy equilibrium and non-consensus fuzzy equilibrium. Therefore, the GMCR technique considering DMs’ consensus can effectively predict the various possible solutions of conflict development under different DMs’ behavior preferences and provide new insights for analysts into a conflict.

Team Production and Esteem: A Dual Selves Model with Belief-Dependent Preferences

We propose a dual selves model to integrate affective responses and belief-dependent emotions into game theory. We apply our model to team production and model a worker as being composed of a rational self, who chooses effort, and an emotional self, who expresses esteem. Similar to psychological game theory, utilities depend on beliefs, but only indirectly. More concretely, emotions affect utilities, and the expression of emotions depends on updated beliefs. Modeling affective responses as actions chosen by the emotional self allows us to apply standard game-theoretic solution concepts. The model reveals that with incomplete information about abilities, workers only choose high effort if esteem is expressed based on interpersonal comparisons and if the preference for esteem is a status preference.

On the Inducibility of Stackelberg Equilibrium for Security Games

Strong Stackelberg equilibrium (SSE) is the standard solution concept of Stackelberg security games. As opposed to the weak Stackelberg equilibrium (WSE), the SSE assumes that the follower breaks ties in favor of the leader and this is widely acknowledged and justified by the assertion that the defender can often induce the attacker to choose a preferred action by making an infinitesimal adjustment to her strategy. Unfortunately, in security games with resource assignment constraints, the assertion might not be valid; it is possible that the defender cannot induce the desired outcome. As a result, many results claimed in the literature may be overly optimistic. To remedy, we first formally define the utility guarantee of a defender strategy and provide examples to show that the utility of SSE can be higher than its utility guarantee. Second, inspired by the analysis of leader’s payoff by Von Stengel and Zamir (2004), we provide the solution concept called the inducible Stackelberg equilibrium (ISE), which owns the highest utility guarantee and always exists. Third, we show the conditions when ISE coincides with SSE and the fact that in general case, SSE can be extremely worse with respect to utility guarantee. Moreover, introducing the ISE does not invalidate existing algorithmic results as the problem of computing an ISE polynomially reduces to that of computing an SSE. We also provide an algorithmic implementation for computing ISE, with which our experiments unveil the empirical advantage of the ISE over the SSE.

Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics

The standard solution concept for stochastic games is Markov perfect equilibrium (MPE); however, its computation becomes intractable as the number of players increases. Instead, we consider mean field equilibrium (MFE) that has been popularized in the recent literature. MFE takes advantage of averaging effects in models with a large number of players. We make three main contributions. First, our main result provides conditions that ensure the uniqueness of an MFE. We believe this uniqueness result is the first of its nature in the class of models we study. Second, we generalize previous MFE existence results. Third, we provide general comparative statics results. We apply our results to dynamic oligopoly models and to heterogeneous agent macroeconomic models commonly used in previous work in economics and operations.

Common belief of weak-dominance rationality in strategic-form games: A qualitative analysis

We study common belief of weak-dominance rationality in strategic-form games with ordinal utilities, employing a qualitative model of beliefs. We characterize two standard solution concepts for such games: the Iterated Deletion of Börgers-dominated Strategies (IDBS) and the Iterated Deletion of Inferior Strategy Profiles (IDIP). We do so by imposing nested restrictions on the doxastic models: namely, the respective epistemic conditions differ in the fact that IDIP requires the truth axiom whereas IDBS does not. Hence, IDIP refines IDBS.

A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models<sup>†</sup>

Pareto optimality and Nash equilibrium are two standard solution concepts for cooperative and non-cooperative games, respectively. At the outset, these concepts are incompatible-see, for instance, [ 7 ] or [ 10 ]. But, on the other hand, there are particular games in which Nash equilibria turn out to be Pareto-optimal [ 1 ], [ 4 ], [ 6 ], [ 18 ], [ 20 ]. A class of these games has been identified in the context of discrete-time potential games [ 13 ]. In this paper we introduce several classes of deterministic and stochastic potential differential games [ 12 ] in which open-loop Nash equilibria are also Pareto optimal.

Conditional Retrospective Voting in Large Elections

We introduce a solution concept in the context of large elections with private information by embedding a model of boundedly rational voters into an otherwise standard equilibrium setting. A retrospective voting equilibrium (RVE) formalizes the idea that voters evaluate alternatives based on past performance. Since counterfactual outcomes are not observed, the sample from which voters learn is potentially biased, leading to systematically biased beliefs in equilibrium. We provide an explicit learning foundation for RVE and contrast it to standard solution concepts in the literature. (JEL D71, D72, D83)

Solution Concepts and Analysis of Spatially Invariant Hybrid Systems: Exploring Zeno and beyond

Abstract: In this paper we consider hybrid systems, consisting of an infinite number of interconnected spatially invariant (identical) hybrid subsystems described using the hybrid inclusions framework. It can be shown that such interconnections can be very useful in, e.g., the modeling of interconnected networked subsystems that use packet-based communication for the exchange of information, including autonomously driving platoons of vehicles. Interestingly however, due to the infinite-dimensionality of the overall interconnected hybrid system, establishing proper definitions of solutions becomes a difficult task as standard solution concepts do not apply to the systems under study since Zeno behavior (an infinite number of jumps in a bounded time interval) is inevitable. Therefore, we introduce an alternative and natural solution concept for this class of systems, allowing solutions to be defined beyond Zeno points. In addition, based on this novel solution concept, we derive Lyapunov-based conditions for a specific, but relevant class of infinite-dimensional hybrid systems, as used for modeling, for instance, networked control systems, that guarantee UGAS (or UGES) or L p -stability of the overall interconnected system.

Berk-Nash Equilibrium: A Framework for Modeling Agents With Misspecified Models

We develop an equilibrium framework that relaxes the standard assumption that people have a correctly specified view of their environment. Each player is characterized by a (possibly misspecified) subjective model, which describes the set of feasible beliefs over payoff-relevant consequences as a function of actions. We introduce the notion of a Berk–Nash equilibrium: Each player follows a strategy that is optimal given her belief, and her belief is restricted to be the best fit among the set of beliefs she considers possible. The notion of best fit is formalized in terms of minimizing the Kullback–Leibler divergence, which is endogenous and depends on the equilibrium strategy profile. Standard solution concepts such as Nash equilibrium and self-confirming equilibrium constitute special cases where players have correctly specified models. We provide a learning foundation for Berk–Nash equilibrium by extending and combining results from the statistics literature on misspecified learning and the economics literature on learning in games.

Empathetic Games

We investigate the effects of players’ mutual empathy, or the interdependence of players’ payoffs, on their behavior and welfare. To this end, we augment the conventional game-theoretic framework by allowing players’ utilities to depend not only on the strategy profile being played, but also on the realized utilities of other players. We call the resulting framework an empathetic game. Generally, in these games players’ utilities are not reducible to preferences over strategies due to novel feedback effects; thus, standard solution concepts, such as the Nash equilibrium, cannot be applied. We extend the notion of a Nash equilibrium to these settings by endowing players with consistent beliefs regarding others’ payoffs. We show that under mild assumptions our solution concept is well defined. We then apply the framework to analyze how mutual empathy affects strategic interactions, and find that it qualitatively changes players’ incentives. First, empathetic games can generate excessive multiplicity of equilibria owing to many supporting beliefs. Second, we uncover a new source of inefficiency, called payoff inefficiency, as players may coordinate their beliefs on socially suboptimal solutions to the interdependent payoff system. Finally, we show that these two features are common in sympathetic (altruistic) games, in which the payoffs of all players depend positively on others’ payoffs; however, payoff inefficiency never occurs in two-player antipathetic games, in which payoffs are negatively related.

Correlated-belief equilibrium

We introduce a new solution concept, called correlated-belief equilibrium. The difference to Nash equilibrium is that, while each player has correct marginal conjectures about each opponent, it is not necessarily the case that these marginal conjectures are independent. Then, we provide an epistemic foundation and we relate correlated-belief equilibrium with standard solution concepts, such as rationalizability, correlated equilibrium and conjectural equilibrium.

Tenable Strategy Blocks and Settled Equilibria

When people interact in familiar settings, social conventions usually develop so that people tend to disregard alternatives outside the convention. For rational players to usually restrict attention to a block of conventional strategies, no player should prefer to deviate from the block when others are likely to act conventionally and rationally inside the block. We explore two set-valued concepts, coarsely and finely tenable blocks, that formalize this notion for finite normal-form games. We then identify settled equilibria, which are Nash equilibria with support in minimal tenable blocks. For a generic class of normal-form games, our coarse and fine concepts are equivalent, and yet they differ from standard solution concepts on open sets of games. We demonstrate the nature and power of the solutions by way of examples. Settled equilibria are closely related to persistent equilibria but are strictly more selective on an open set of games. With fine tenability, we obtain invariance under the insertion of a subgame with a unique totally mixed payoff-equivalent equilibrium, a property that other related concepts have not satisfied.

An Equilibrium Framework for Players with Misspecified Models

We introduce an equilibrium framework that relaxes the standard assumption that people have a correctly-specified view of their environment. Players repeatedly play a simultaneous-move game where they potentially face both strategic and payoff uncertainty. Each player has a potentially misspecified view of the environment and uses Bayes' rule to update her views based on the (possibly partial) feedback obtained at the end of each period. We show that steady-state behavior of this multi-player decision and learning problem is captured by a generalized notion of equilibrium: a strategy profile such that each player optimizes given certain beliefs and where these beliefs put probability one on those subjective distributions over consequences that are closest in terms of relative entropy to the correct, equilibrium distribution. Standard solution concepts such as Nash equilibrium and self-confirming equilibrium constitute special cases where players learn with correctly-specified models. The framework provides a systematic approach to modeling players with misspecified views and also unifies a specific bounded rationality literature where mistakes are driven by misspecications.