Simultaneous-move Game Research Articles

A Refinement of Logit Quantal Response Equilibrium

Unlike the Nash equilibrium, logit quantal response equilibrium is affected by positive affine transformations of players’ von Neumann–Morgenstern utility payoffs. This paper presents a modification of a logit quantal response equilibrium that makes this equilibrium solution concept invariant to arbitrary normalization of utility payoffs. Our proposed modification can be viewed as a refinement of logit quantal response equilibria: instead of obtaining a continuum of equilibria (for different positive affine transformations of utility function) we now obtain only one equilibrium for all possible positive affine transformations of utility function. We define our refinement for simultaneous-move noncooperative games in the normal form.

A model of terrorism and counterterrorism with location choices

We study a model of terrorism and counterterrorism with three geographic target locations, in which the last location is chosen by the terrorists and is unknown to the defender. We find the choice of location and defender’s allocation of defensive resources to depend on whether the first two locations are attacked simultaneously or sequentially. In the sequential-move game, the final attack is closer to the low-valued target than it is in the simultaneous-move game. In addition, the defender chooses to defend only the location that is attacked first in the sequential game. Results from the model also can shed light on less provocative criminal events such as bank robberies and drug dealings.

Equilibrium vaccination patterns in incomplete and heterogeneous networks

Under-vaccination is a usual concern of disease control studies, but this paper employs a simultaneous-move game in a three-agent contagion network to show that it is only one of the three inefficient patterns. When the network structure is incomplete or individual characteristics are heterogeneous, there exist new types of Nash equilibrium outcomes with either the right number but wrong set of people getting vaccinated or too many vaccinations, and these equilibria are robust to standard refinements. While untargeted policies can correct the standard under-vaccination problem, targeted policies are more palatable for correcting the new inefficiencies. Universal mandates can never improve on any Nash equilibria.

Endogenous Timing and Income Inequality in the Voluntary Provision of Public Goods: Theory and Experiment

This paper investigates how the heterogenous incomes and preferences of potential donors affect the timing of contribution decisions when it is endogenously determined by contributors themselves. More specifically, we use a simple setting with two donors, Cobb-Douglas preferences, and complete information to investigate how income inequality affects the endogenous choices of contribution timing and the level of the voluntary supplied public goods. This paper obtains the following results. First, when income is extremely unequal, potential contributors are indifferent between the timing choices of simultaneous and sequential moves, even if they have different preferences towards a public good. Second, as income inequality decreases, the simultaneous move-game is increasingly likely to emerge, because all potential contributors prefer to act as a leader. Third, in the presence of multiple public goods, contributors with higher valuations for one public good tend to be first contributors to that one. Fourth, these theoretical predictions regarding the timing decisions of individuals are not supported by the laboratory experiment, although those regarding individuals’ contribution decisions are consistent with the experimental results.

Modified Benders Decomposition for Solving Transmission Investment Game With Risk Measure

This paper proposes a mathematical model for transmission investment game where there are several transmission planners (TPs). The model is developed assuming a simultaneous-move game between TPs. Each TP maximizes the total surplus (producers’, consumers’, and transmission surplus) minus the investment cost of its region given the investment decisions of rival TPs. The transmission investment risk is also considered using the probability-of-shortfall measure. We assume one market operator who dispatches the generators in all TP's regions. The risk-constrained Nash equilibria model is formulated as a mixed-integer linear program (MILP). To solve the proposed MILP, a solution algorithm is proposed that combines the standard branch-and-bound algorithm (BB) with a proposed modified benders decomposition algorithm (MBD). The proposed BB-MBD algorithm is also parallelized to improve the computation performance. To improve the coordination between TPs, a supporting budget mechanism is also mathematically modeled in the MILP. The numerical results are carried out using the 9-bus 3-area and the IEEE Three Area RTS-96 networks. The computational performance of proposed BB-MBD is compared with standard BB algorithm.

Choosing an online retail channel for a manufacturer: Direct sales or consignment?

We analyze a manufacturer's e-channel decision problem in which the manufacturer selects a direct-sales channel or a third-party consignment channel to complement his existing physical retail channel. We accordingly investigate two possible dual channels: a PD system involving a physical channel and a direct e-channel, and a PC system consisting of a physical channel and a consignment e-channel. For each system, we examine both a sequential-move game and a simultaneous-move game, as the manufacturer can strategically decide to announce his pricing decision before the physical retailer or to make his pricing decision with no communication with the physical retailer. Our analytical results indicate that, if the manufacturer's unit operating cost in the direct e-channel or the e-tailer's revenue allocation ratio in the consignment e-channel is sufficiently small, then the manufacturer has an incentive to adopt an e-channel. The manufacturer can always gain a higher profit by announcing his pricing decision before the physical retailer. If the manufacturer aims at increasing the demand, then he may choose the simultaneous-move game. Moreover, when the manufacturer selects an e-channel to increase his profit, he should adopt a direct e-channel if his unit e-channel operating cost is below a certain threshold that is dependent on the e-tailer's revenue allocation ratio, and adopt a consignment e-channel otherwise. A similar managerial insight is drawn when the manufacturer intends to increase the demand.

Predicting human behavior in unrepeated, simultaneous-move games

It is commonly assumed that agents will adopt Nash equilibrium strategies; however, experimental studies have demonstrated that this is often a poor description of human players' behavior in unrepeated normal-form games. We analyze five widely studied models of human behavior: Quantal Response Equilibrium, Level-k, Cognitive Hierarchy, QLk, and Noisy Introspection. We performed what we believe is the most comprehensive meta-analysis of these models, leveraging ten datasets from the literature recording human play of two-player games. We first evaluated predictive performance, asking how well each model fits unseen test data using parameters calibrated from separate training data. The QLk model (Stahl and Wilson, 1994) consistently achieved the best performance. Using a Bayesian analysis, we found that QLk's estimated parameter values were not consistent with their intended economic interpretations. Finally, we evaluated model variants similar to QLk, identifying one (Camerer et al., 2016) that achieves better predictive performance with fewer parameters.

Modeling a Multitarget Attacker–Defender Game with Budget Constraints

Though the choices of terrorists’ attack targets are vast, their resources are limited. In this paper, a game-theoretical model is proposed to study both the defender’s (government) and the attacker’s (terrorist) expenditures among multiple targets under budget constraints to guide investment in defense. We study how the defender’s and the attacker’s equilibrium allocations depend on the budget constraints, target valuations, cost effectiveness of their investments, and inherent defense levels of targets in both sequential-move and simultaneous-move games. The equilibrium solutions are provided using the Karush–Kuhn–Tucker conditions. At the subgame-perfect Nash equilibrium, the defender’s total marginal effects are the same among targets. Moreover, the defender’s total marginal effects can be decomposed into direct and indirect effects. We also use the multiple-infrastructure and multiple-urban-area data sets to demonstrate the model results. The regression analysis shows that both the attacker’s and the defender’s equilibrium investments increase with their own target valuations, because a higher valuation for themselves means a higher attractiveness. Interestingly, the attacker’s equilibrium investment is negatively correlated with the defender’s target valuations, since a higher defender’s valuation would make it more difficult for the attacker to successfully attack the target. By contrast, the defender’s equilibrium investment is positively correlated with the attacker’s target valuations, as a higher attacker’s valuations would increase the urgency for the defender to protect the target. To show the utility of the new model, we compare the results of this model and the model in which the defender assumes that only a single target will be attacked when there could actually be multiple targets attacked. Our results show that the defender will suffer higher expected losses if she assumes that the attacker will attack only one target. The analysis of the attacker’s and the defender’s budget constraints show that (a) the higher the budget the defender has, the less likely it is that her most valuable target will be attacked; (b) a higher proportion of defense resources should be allocated to the most valuable target if the defender’s budget is low; and (c) the attacker is less concentrated on attacking the most valuable target and spreads the resources to attack more targets as his budget increases.

Costly Leader Games with a Probabilistically Non-Strategic Leader

I consider a two-person costly leader game — in which the follower endogenously chooses whether to buy information about the leader, and a follow-up action — with a twist. With a known probability, the leader is a nonstrategic type. A strategic type leader may choose to masquerade as a nonstrategic type, at a cost. I show that, if the follower’s cost of information is not too large, and the probability of the leader being nonstrategic is neither too large nor too small, this game has no pure strategy equilibrium. Moreover, the equilibrium of the simultaneous-move complete information game is inaccessible as follower information cost converges to zero. There is no equilibrium outcome in which leader advantage is destroyed: however, a mixed strategy equilibrium exists which does preserve leader advantage in the sense that payoffs and strategies converge to those of the sequential-move complete information equilibrium as information cost tends to zero. My results differ from some traditional results in costly leader games, and are due to the interaction of two forces, the “type uncertainty” and the “money down the drain” effects. To my knowledge this is the first paper to integrate behavioral types into costly leader games (other papers considering heterogeneity in type do not consider nonstrategic players).

Game Theory-Based Control System Algorithms with Real-Time Reinforcement Learning: How to Solve Multiplayer Games Online

Complex human-engineered systems involve an interconnection of multiple decision makers (or agents) whose collective behavior depends on a compilation of local decisions that are based on partial information about each other and the state of the environment [1]-[4]. Strategic interactions among agents in these systems can be modeled as a multiplayer simultaneous-move game [5]-[8]. The agents involved can have conflicting objectives, and it is natural to make decisions based upon optimizing individual payoffs or costs.

On the Two-User Multi-Carrier Joint Channel Selection and Power Control Game

In this paper, we propose a hierarchical game approach to model the energy efficiency maximization problem where transmitters individually choose their channel assignment and power control. We conduct a thorough analysis of the existence, uniqueness and characterization of the Stackelberg equilibrium. Interestingly, we formally show that a spectrum orthogonalization naturally occurs when users decide sequentially about their transmitting carriers and powers, delivering a binary channel assignment. Both analytical and simulation results are provided for assessing and improving the performances in terms of energy efficiency and spectrum utilization between the simultaneous-move game (with synchronous decision makers), the social welfare (in a centralized manner) and the proposed Stackelberg (hierarchical) game. For the first time, we provide tight closed-form bounds on the spectral efficiency of such a model, including correlation across carriers and users. We show that the spectrum orthogonalization capability induced by the proposed hierarchical game model enables the wireless network to achieve the spectral efficiency improvement while still enjoying a high energy efficiency.

Sequential Coordination of Transmission Expansion Planning With Strategic Generation Investments

This paper proposes mathematical models for sequential coordination of transmission expansion planning with strategic generation investments. The proactive and reactive coordinations are modeled and studied. The interaction between transmission company (Transco) and strategic generation companies (Gencos) is modeled using the sequential-move game. This is while the interaction between the strategic Gencos is modeled as a simultaneous-move game. In the proactive coordination, the Transco expands its future transmission capacities taking into account the strategic investments by Gencos. In the reactive coordination, strategic Gencos move first and expand their future generation capacities and then Transco expands the transmission capacity. The proactive coordination is modeled as a mixed-integer bilevel linear program (MIBLP) and the reactive coordination is modeled as a mixed-integer linear program (MILP). The MIBLP has binary variables in both upper and lower levels. The Moore–Bard algorithm is parallelized and used to solve the MIBLP. The mathematical models and the parallelized Moore–Bard algorithm are tested on 3-bus and 6-bus example systems and the modified IEEE-RTS96. Also, the IEEE 118-bus test system is studied using a heuristic version of the Moore–Bard algorithm.

Modularity, Lead time and Return Policy for Supply Chain in Mass Customization System

AbstractMass Customization (MC) is a flexible manufacturing system with features of Mass Production (MP) and Customization Production (CP). However, there is few researches about competition & cooperation between the upstream MP firm (module manufacturer) and downstream CP firm (assembler) under MC supply chain scenario. From supply chain perspective, this paper first develops the base models considering the influences of return policy, modularity level, production lead time and pricing factors. Furthermore, according to the different decision-making situations, three kinds of MC supply chain models in competitive or cooperative environment (i.e. simultaneous-move game, sequential-move game and the cooperative game) have been built, then, the optimal solution of each model have been analyzed and compared, and coordination mechanism is design to cooperate in MC supply chain via profit-sharing with Nash bargaining power. Finally, through the numerical analysis, we find the highest profit is from the coopera...

Solving the Volunteer's Dilemma: The Efficiency of Rewards versus Punishments

This paper studies a class of second-best solutions to the Volunteer's Dilemma. We consider a simultaneous-move game involving n players, each of whom can prevent social harm at a cost. Players could be rewarded for helping, be punished for not helping, or be subject to any combination of rewards and punishments. One such mechanism, for example, offers each player a reward for helping and imposes on each player a punishment for not helping if no other player provides help. We show that, because players' efforts are substitutes, the use of rewards, rather than punishments, minimizes social costs (sum of expected cost of helping, expected harm, and expected costs of implementation) for nearly any number of players and ratio of cost-of-helping to social harm. Our analysis helps to explain the paucity of affirmative legal duties and is consistent with pecuniary rewards bestowed on volunteers such as whistle-blower and most-wanted awards.

Key leaders in social networks

This paper examines optimal targeting and sequencing strategies in the setup proposed by Ballester et al. [3]. The setup features payoff externalities and strategic complementarity among players, who non-cooperatively determine their contributions. We first analyze a two-stage game in which players in the leader group make contributions before the follower group. We construct an exact index to identify the (single) key leader, and demonstrate that the key leader can differ substantially from the key player who most influences the network in the simultaneous-move game. Using Taylor expansions on the strength of network effects, we establish an isomorphism between the optimal leader group selection (targeting) strategy and the classical weighted maximum-cut problem. This approach leads to some design principles for unweighted complete graphs and bipartite graphs.We then allow for an arbitrary sequence of players' moves. If we refine any sequence by making some groups of simultaneous movers act in sequence, their aggregate contribution increases. Consequently, the optimal sequence is a chain structure. When players have homogeneous intrinsic valuations, any chain, arbitrarily ordered, maximizes the aggregate contribution. We further consider the possibility of shrinking the number of stages needed for the optimal sequence. In general, our results hold when the network game exhibits strategic substitutes.

The Benefit of Sequentiality in Social Networks

This paper examines optimal targeting and sequencing strategies in the setup proposed by Ballester et al. (2006). The setup features payoff externalities and strategic complementarity among players, who non-cooperatively determine their contributions. We first analyze a two-stage game in which players in the leader group make contributions before the follower group. We construct an exact index to identify the (single) key leader, and demonstrate that the key leader can differ substantially from the key player who most influences the network in the simultaneous-move game. Using Taylor expansions on the strength of network effects, we establish an isomorphism between the optimal leader group selection (targeting) strategy and the classical weighted maximum-cut problem. This approach leads to some design principles for unweighted complete graphs and bipartite graphs.We then allow for an arbitrary sequence of players' moves. If we refine any sequence by making some groups of simultaneous movers act in sequence, their aggregate contribution increases. Consequently, the optimal sequence is a chain structure. When players have homogeneous intrinsic valuations, any chain, arbitrarily ordered, maximizes the aggregate contribution. We further consider the possibility of shrinking the number of stages needed for the optimal sequence. In general, our results hold when the network game exhibits strategic substitutes.

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.

Indirect reciprocity in three types of social dilemmas

Indirect reciprocity is a key mechanism for the evolution of human cooperation. Previous studies explored indirect reciprocity in the so-called donation game, a special class of Prisoner׳s Dilemma (PD) with unilateral decision making. A more general class of social dilemmas includes Snowdrift (SG), Stag Hunt (SH), and PD games, where two players perform actions simultaneously. In these simultaneous-move games, moral assessments need to be more complex; for example, how should we evaluate defection against an ill-reputed, but now cooperative, player? We examined indirect reciprocity in the three social dilemmas and identified twelve successful social norms for moral assessments. These successful norms have different principles in different dilemmas for suppressing cheaters. To suppress defectors, any defection against good players is prohibited in SG and PD, whereas defection against good players may be allowed in SH. To suppress unconditional cooperators, who help anyone and thereby indirectly contribute to jeopardizing indirect reciprocity, we found two mechanisms: indiscrimination between actions toward bad players (feasible in SG and PD) or punishment for cooperation with bad players (effective in any social dilemma). Moreover, we discovered that social norms that unfairly favor reciprocators enhance robustness of cooperation in SH, whereby reciprocators never lose their good reputation.

Cooperative advertising with a strong retailer and a weak manufacturer: a closer look at the consumer electronics industry

Models of cooperative advertising analysed until recently have mainly revolved around two cases: 1) strong manufacturer, weak retailer; 2) strong manufacturer, equally strong retailer. These are not generalisable to market dynamics in all countries and all industries. Despite being understudied, a case that is growing in significance is that of weak manufacturers and strong retailers. In this study, we focus on a manufacturer that is excessively weak in comparison to the retailer. Such manufacturers are dominating the consumer electronics industry in developing countries where there is a need for low-priced products with acceptable quality. They partner with strong retailers, and invest heavily in cooperative advertising in order to communicate product related information to potential customers and generate sales. We aim to analyse the weak manufacturer-strong retailer setting with a game theoretic approach, focusing on the strategic impacts of retail channel marketing and co-op advertising sharing policy. Sequential and simultaneous-move game models are examined to determine the optimal co-op advertising strategies that maximise both players’ profits. It is found that when the distribution of power between the retailer and the manufacturer is highly unbalanced, the only feasible non-cooperative game model is the leader-follower scheme in which the retailer decides both the channel marketing investment level and the co-op advertising sharing policy.

An Application of Decision Theoretical Models of Probabilistic Choice to Solving 2x2 Simultaneous-Move Noncooperative Games in the Normal Form

Several models of probabilistic choice under uncertainty allow for deterministic choice if one act state-wise dominates the other. Such models have a natural application in game theory where probabilistic choice corresponds to mixed strategies, deterministic choice — to pure strategies and state-wise dominance — to strategic dominance. This paper considers an application to the simplest possible 2x2 simultaneous-move noncooperative game in the normal form. We derive a new equilibrium solution concept that coincides with the standard Nash equilibrium in pure strategies but resembles a quantal response equilibrium in mixed strategies. In particular, players randomize between strategies not in order to keep their opponent indifferent between his or her strategies (as in the mixed strategy Nash equilibrium) but because they are more likely to (but not always) choose strategies yielding higher utility.