Payoff-dominant Equilibrium Research Articles

Evolutionary dynamics in bilingual games

In two-strategy coordination games with distinct payoff- and risk-dominant equilibria, existing results show that the inefficient risk-dominant equilibrium is uniquely selected under many evolutionary dynamics. In the above class of coordination games, we study the effect of introducing a bilingual strategy that is compatible with both of the existing strategies. An agent playing the bilingual strategy incurs an additional adoption cost but never miscoordinates with any other agent. We show that if the adoption cost of the bilingual strategy is low, then the efficient payoff-dominant equilibrium can be uniquely selected under many evolutionary dynamics.

Transitions between equilibria in Bilingual Games under Probit Choice

We study the effect of introducing a bilingual strategy on the long run equilibrium outcome in a class of two-strategy coordination games with distinct payoff- and risk-dominant equilibria under the probit choice rule. Existing results show that in the class of two-strategy games under consideration, the inefficient risk dominant equilibrium is selected in the long run under noisy best response models. We show that if the cost of the bilingual option is sufficiently low then the efficient payoff dominant equilibrium will be selected in the long run under the probit choice rule.

Evolutionary games on multilayer networks: coordination and equilibrium selection

We study mechanisms of synchronisation, coordination, and equilibrium selection in two-player coordination games on multilayer networks. We investigate three possible update rules: the replicator dynamics (RD), the best response (BR), and the unconditional imitation (UI). Players interact on a two-layer random regular network. The population on each layer plays a different game, with layer I preferring the opposite strategy to layer II. We measure the difference between the two games played on the layers by a difference in payoffs, and the inter-connectedness by a node overlap parameter. We discover a critical value of the overlap below which layers do not synchronise, i.e. they display different levels of coordination. Above this threshold both layers typically coordinate on the same strategy. Surprisingly, there is a symmetry breaking in the selection of equilibrium—for RD and UI there is a phase where only the payoff-dominant equilibrium is selected. It is not observed, however, for BR update rule. Our work is an example of previously observed differences between the update rules. Nonetheless, we took a novel approach with the game being played on two inter-connected layers. As we show, the multilayer structure enhances the abundance of the Pareto-optimal equilibrium in coordination games with imitative update rules.

Network coevolution drives segregation and enhances Pareto optimal equilibrium selection in coordination games

In this work we assess the role played by the dynamical adaptation of the interactions network, among agents playing Coordination Games, in reaching global coordination and in the equilibrium selection. Specifically, we analyze a coevolution model that couples the changes in agents’ actions with the network dynamics, so that while agents play the game, they are able to sever some of their current connections and connect with others. We focus on two action update rules: Replicator Dynamics (RD) and Unconditional Imitation (UI), and we define a coevolution rule in which, apart from action updates, with a certain rewiring probability p, agents unsatisfied with their current connections are able to eliminate a link and connect with a randomly chosen neighbor. We call this probability to rewire links the ‘network plasticity’. We investigate a Pure Coordination Game (PCG), in which choices are equivalent, and on a General Coordination Game (GCG), for which there is a risk-dominant action and a payoff-dominant one. Changing the plasticity parameter, there is a transition from a regime in which the system fully coordinates on a single connected component to a regime in which the system fragments in two connected components, each one coordinated on a different action (either if both actions are equivalent or not). The nature of this fragmentation transition is different for different update rules. Second, we find that both for RD and UI in a GCG, there is a regime of intermediate values of plasticity, before the fragmentation transition, for which the system is able to fully coordinate on a single component network on the payoff-dominant action, i.e., coevolution enhances payoff-dominant equilibrium selection for both update rules.

Discontinuous and continuous stochastic choice and coordination in the lab

We experimentally test theoretical predictions on equilibrium selection in a two-player coordination (investment) game. Through a minimal visual variation, our design prompts participants to play strategies whereby investing probability is either continuous or discontinuous in the payoff-relevant state. When participants use continuous strategies, average behavior is consistent with play in the risk-dominant equilibrium, the unique theoretical prediction. When they use discontinuous strategies, average behavior is closer to the payoff-dominant equilibrium strategy. In this case, the theory predicts multiple equilibria, for which we find no support. Additionally, we extend the theory to heterogeneous populations: the set of equilibria monotonically decreases in the proportion of players who use continuous strategies.

Quadratic Privacy-Signaling Games and the MMSE Information Bottleneck Problem for Gaussian Sources

We investigate a privacy-signaling game problem in which a sender with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The sender aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework with quadratic costs where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equilibrium among all admissible policies is attained by a set of explicitly characterized linear policies. We also show that a payoff dominant Nash equilibrium coincides with a Stackelberg equilibrium. We formulate the information bottleneck problem within our Stackelberg framework under the mean squared error distortion criterion where the information bottleneck setup has a further restriction that only one of the random variables is observed at the sender. We show that this MMSE Gaussian Information Bottleneck Problem admits a linear solution which is explicitly characterized in the paper. We provide explicit conditions on when the optimal solutions, or equilibrium solutions in the Nash setup, are informative or noninformative.

Can two competing on-demand service platforms be profitable?

Can two competing on-demand service platforms be profitable in equilibrium? This question is well-studied for firms competing purely on price in a single-sided market, but it is not well-understood for competing service platforms that use price to influence passenger demand and wage to influence driver participation in a “two-sided” market. To fill this research gap, we analyze the equilibrium structure of different variants of a 2-stage non-cooperative game in which both platforms use lower prices and waiting time to compete for more customers and higher wages and utilization to entice more providers to participate.To explicate our analysis and isolate the effect of each element, we first examine a parsimonious and stylized model that is based on the case when both firms operate under seven operational assumptions: (1) Non-exclusive providers; (2) Non-exclusive customers; (3) Pure pricing strategies; (4) Constant pricing; (5) Homogeneous services; (6) Homogeneous providers; and (7) Homogeneous customers. We find that only one platform can sustain in a payoff dominant stable equilibrium. Then we examine whether this “winner-take-all” equilibrium would persist when those operational environmental assumptions are relaxed separately. Our analysis reveals that the “winner-take-all” phenomenon continues to persist under promotional pricing strategies, service differentiation, and heterogeneous service providers. Interestingly, we find both platforms can sustain profitably when each platform engages providers (or customers) exclusively, when platforms adopt time-based pricing strategies, or when customers are heterogeneous. Our results offer insights into different operating environments under which both platforms can both sustain profitably in equilibrium.

Coordination and equilibrium selection in games: the role of local effects

We study the role of local effects and finite size effects in reaching coordination and in equilibrium selection in two-player coordination games. We investigate three update rules — the replicator dynamics (RD), the best response (BR), and the unconditional imitation (UI). For the pure coordination game with two equivalent strategies we find a transition from a disordered state to coordination for a critical value of connectivity. The transition is system-size-independent for the BR and RD update rules. For the IU it is system-size-dependent, but coordination can always be reached below the connectivity of a complete graph. We also consider the general coordination game which covers a range of games, such as the stag hunt. For these games there is a payoff-dominant strategy and a risk-dominant strategy with associated states of equilibrium coordination. We analyse equilibrium selection analytically and numerically. For the RD and BR update rules mean-field predictions agree with simulations and the risk-dominant strategy is evolutionary favoured independently of local effects. When players use the unconditional imitation, however, we observe coordination in the payoff-dominant strategy. Surprisingly, the selection of pay-off dominant equilibrium only occurs below a critical value of the network connectivity and disappears in complete graphs. As we show, it is a combination of local effects and update rule that allows for coordination on the payoff-dominant strategy.

Identifying the Reasons for Coordination Failure in a Laboratory Experiment

We investigate the effect of absence of common knowledge on the outcomes of coordination games in a laboratory experiment. Using cognitive types, we can explain coordination failure in pure coordination games while differentiating between coordination failure due to first- and higher-order beliefs. In our experiment, around 76% of the subjects have chosen the payoff-dominant equilibrium strategy despite the absence of common knowledge. However, 9% of the players had first-order beliefs that lead to coordination failure and another 9% exhibited coordination failure due to higher-order beliefs. Furthermore, we compare our results with predictions of commonly used models of higher-order beliefs.

Strategic Leaks in First-Price Auctions and Tacit Collusion: The Case of Spying and Counter-Spying

We analyze strategic leaks due to spying out a rival’s bid in a first-price auction. Such leaks induce sequential bidding, complicated by the fact that the spy may be a counterspy who serves the interests of the spied at bidder and reports strategically distorted information. This ambiguity about the type of spy gives rise to a non-standard signaling problem where both sender and receiver of messages have private information and the sender has a chance to make an unobserved move. Whereas spying without counterspy exclusively benefits the spying bidder, the potential presence of a counterspy yields a collusive outcome, even if the likelihood that the spy is a counterspy is arbitrarily small. That collusive impact shows up in all equilibria and is strongest in the unique pooling equilibrium which is also the payoff dominant equilibrium.

Starting small to communicate

We analyze a repeated cheap-talk game in which the receiver is privately informed about the conflict of interest between herself and the sender and either the sender or the receiver controls the stakes involved in their relationship. We focus on payoff-dominant equilibria that satisfy a Markovian property and show that if the potential conflict of interest is large, then the stakes increase over time, i.e., “starting small” is the unique equilibrium arrangement. In each period, the receiver plays the sender's ideal action with positive probability and the sender provides full information as long as he has always observed his ideal actions in the past. We also show that as the potential conflict of interest increases, the extent to which the stakes are back-loaded increases, i.e., stakes are initially smaller but grow faster.

Transitions between equilibria in bilingual games under logit choice

We study the effect of introducing a bilingual option on the long run equilibrium outcome in a class of two-strategy coordination games with distinct payoff and risk dominant equilibria under the logit choice rule. Existing results show that in the class of two-strategy games under consideration, the inefficient risk dominant equilibrium is selected in the long run under noisy best response models. We show that if the cost of the bilingual option is sufficiently low then the efficient payoff dominant equilibrium will be selected in the long run under the logit choice rule.

Coordination Games in Continuous Time

This paper studies the impact of continuous time interaction on two iconic coordination games: stag hunt and battle of the sexes in a laboratory environment and compare results to possible theoretical explanations. Experimental results show that subjects consistently coordinate better in continuous time than in discrete time under various treatment environments. In continuous time, they are also more likely to converge to payoff-dominant equilibrium in stag hunt games and alternating dynamics in battle of the sexes games. Furthermore, the coordination rate is affected by complexity of action sets and weakly influenced by payoff matrices. The paper also explores some stylized facts that result in the treatment effects.

Can Two Competing On-Demand Service Platforms be Both Profitable?

Problem Definition: Can two competing on-demand service platforms be profitable in equilibrium? The question is well-studied for firms competing purely on price in a single-sided market, but it is not well-understood for on-demand service platforms competing on price and wage in a “two-sided” market. Academic/Practical Relevance: As entrepreneurs develop and as venture capital firms finance various on-demand service platforms, it is important for them to examine whether multiple competing platforms can co-exist profitably. Methodology: In this paper, we analyze the equilibrium structure by solving different variants of a 2-stage non-cooperative game in which both platforms use lower prices and waiting time to compete for more customers and higher wages and utilization to entice more providers to participate. Results: We first examine a base case when both firms operate under seven operational assumptions: 1) Non-exclusive providers; 2) Non-exclusive customers; 3) Pure pricing strategies; 4) Constant pricing; 5) Homogeneous services; 6) Homogeneous providers; and 7) Homogeneous customers. We find that only one platform can sustain in a payoff dominant stable equilibrium. We examine whether this “winner-take-all” equilibrium would persist when those operational environmental assumptions are relaxed separately. Our analysis reveals that the “winner-take-all” phenomenon continues to persist under promotional pricing strategies, service differentiation, and heterogeneous service providers. However, both platforms can sustain profitably when each platform engages providers (or customers) exclusively, when platforms adopt time-based pricing strategies, or when customers are heterogeneous. Managerial Implications: Our results offer insights into different operating environments under which both platforms can both sustain profitably in equilibrium.

Spatial externalities and risk in interdependent security games

Individuals regularly invest in self-protection to reduce the risk of an adverse event. The effectiveness of self-protection often depends on the actions of other economic agents and can be modeled as a stochastic coordination game with multiple Pareto-ranked equilibria. We use lab experiments to analyze tacit coordination in stochastic games with two kinds of interdependencies in payoffs: “non-spatial” in which every agent’s action has an impact on the risk faced by every other agent, and “spatial” in which agents only impact the risk faced by their immediate neighbors. We also compare behavior in the stochastic games to deterministic versions of the same games. We find that coordination on the payoff-dominant equilibrium is significantly easier in the deterministic games than in the stochastic games and that spatial interdependencies lead to greater levels of coordination in the deterministic game but not in the stochastic game. The difficulty with coordination observed in the stochastic games has important implications for many real-world examples of interdependent security and also illustrates the importance of not relying on data from deterministic experiments to analyze behavior in settings with risk.

Green Investment and Coordination Failure: An Investors' Perspective

To achieve the goal of keeping global warming well below 2 °C, private investors have to shift capital from brown to green infrastructures and technologies and provide additional green investment. In this paper, we present a game-theoretic perspective on the challenge of triggering such investments. The question of climate change mitigation is often related to the prisoner's dilemma, a game with one Nash equilibrium. However, the authors perceive investment for mitigation and adaptation as a coordination problem of selecting among multiple equilibria. To illustrate this, we model a non-cooperative coordination game, related to the stag hunt, with a brown equilibrium with lower payoffs that can be achieved single-handedly and a green equilibrium with higher payoffs that requires coordination. As multiple experiments show, in such games actors often fail to coordinate on a payoff dominant equilibrium due to uncertainty. Thus, we discuss how uncertainty could be reduced along two options: one that concerns a change in the payoff structure of the game and another that concerns subjective probabilities.

A NonCooperative Approach to Cost Allocation in Joint Replenishment

We consider the infinite-horizon multiple retailer joint replenishment problem with first-order interaction. In this model, the joint setup cost incurred by a group of retailers placing an order simultaneously consists of a group-independent major setup cost and retailer-specific minor setup costs. The goal is to determine an inventory replenishment policy that minimizes the long-run average system-wide cost. In this paper, we adopt a noncooperative approach to study the joint replenishment game. We consider the allocation rule in which the major setup cost is split equally among the retailers who place an order together, and each retailer pays his own holding and minor setup costs. Given the preannounced allocation rule, each retailer determines his replenishment policy to minimize his own cost anticipating the other retailers’ strategy. We show that a payoff dominant Nash equilibrium exists, and quantify the efficiency loss of the noncooperative outcome relative to the social optimum. Although the worst-case ratio between the best decentralized outcome and the social optimum is O((ln n)1/2), where n is the number of retailers, numerical results suggest that the best equilibrium is near optimal. The online appendix is available at https://doi.org/10.1287/opre.2017.1645 .

Coordination and transfer

We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.

Information transparency and equilibrium selection in coordination games: an experimental study

We experimentally investigate the role of information transparency for equilibrium selection in stag hunt coordination games. These games can be transformed from a prisoner’s dilemma game by introducing a centralized reward or punishment scheme. We aim to explore the impact of the disclosure of information on how final payoffs are derived on players’ incentive to coordinate on the payoff-dominant equilibrium. We find that such information disclosure significantly increases the tendency of players to play the payoff-dominant strategy and reduces the occurrence of coordination failure. The mechanism works directly through the positive impact of disclosure on the saliency of the payoff-dominant equilibrium, and indirectly through the positive influence of disclosure on players’ belief about the likelihood of cooperation by the opponent.

Costly Pre-Play Communication and Coordination in Stag-Hunt Games

In this paper, we experimentally investigate the impact of costly indirect and direct messages on coordination levels in a stag-hunt game. We also compare the coordination rates with costly pre-play communication to the rates with costless pre-play communication. Three main insights emerge from our experiments. First, we find a significant decrease in message usage with message cost in both treatments and a higher decrease in the indirect-message treatment. Second, we find that although there is no significant effect of costless or costly indirect messages on the frequency of risky actions, both costless and costly direct messages significantly increase the frequency of risky actions. Third, while we find a significant increase in the coordination rate on the payoff-dominant equilibrium from costless indirect message treatment to costly indirect message treatment, this rate significantly decreases from costless direct message to costly direct message treatment. Our findings show that depending on the structure of messages, message cost may increase or decrease the coordination rates on the payoff-dominant equilibrium.