A mixed behaviour equilibrium model with mode choice and its application to the endogenous demand of automated vehicles

This study develops a mixed behavioural equilibrium model with explicit consideration of mode choice (MBE-MC) in a transportation system where fully automated vehicles (AV) coexist with conventional human-driven vehicles (HV). For the mode choice, travellers select among three options, following a logit modal split: driving their private HV, or taking an AV mobility service provided by either a firm or the government. For the route choice, the HV drivers follow the random utility maximisation principle while central agents route the AV passengers following the Cournot Nash (firm agent) or Social Optimal (government agent) principles. We consider two types of travel costs (i.e. travel time and monetary travel cost) to characterise the new features (e.g. expanded link capacity and reduced value of time) of the mixed AV–HV transportation system. We model the MBE-MC problem in a combined mode–route choice framework and formulate it as a route-based variational inequality (VI) problem. We show the equivalence between the VI formulation and the MBE-MC problem, and the existence of a solution to the MBE-MC problem. Then, we modify a partial linearisation algorithm for solving the proposed model. Numerical results validate the equilibrium conditions and show the efficacy of the new model in capturing the features of the mixed AV–HV transportation system. The impact patterns of different parameters on (1) the network performance in terms of AV share and system cost and (2) on the solution efficiency are analysed.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Optimal information exchange in contests

We study optimal exchange of private information in a two-player all-pay auction contest with independent private binary values. A benevolent information center who is informed about the players’ values facilitates the exchange of information by disclosing a signal publicly. The informativeness of the signal determines the monotonicity of the unique symmetric equilibrium and the players’ expected payoff. We characterize the upper bound of players’ expected payoff and the corresponding optimal signals utilizing such a relation between the informativeness and the payoff. When the players are ex ante sufficiently heterogeneous, the optimal signals work through an information-rent channel by inducing allocative efficient contests. When the players are ex ante sufficiently homogeneous, the optimal signals work through an unlevel-playing-field channel by inducing asymmetric contests. In order to guarantee efficient allocation, a regulator can punish any exchange of information when the players are sufficiently homogeneous and impose no restrictions when they are sufficiently heterogeneous.     

Stopping with congestion and private payoffs

This paper analyses a two-player stopping game with multiarmed bandits in which each player chooses between learning about the quality of her private risky arm and competing for the use of a single shared safe arm. The qualities of the players’ risky arms are independent. A player whose risky arm produces a success no longer competes for the safe arm. We assume that a player observes her opponent’s actions but not his realised payoffs. She is therefore never certain whether her opponent is still competing for the safe arm. When the players’ prior probabilities of success are sufficiently close, there exists no pure strategy equilibrium, and we characterise the unique mixed strategy equilibrium. Otherwise, the unique equilibrium is in pure strategies. The amount of experimentation performed in equilibrium is inefficiently low but, for many priors, higher than if successes are publicly observed.     

The generalized Stackelberg equilibrium of the all-pay auction with complete information

We characterize the equilibrium of the all-pay auction with general convex cost of effort and sequential effort choices. We consider a set of n players who are arbitrarily partitioned into a group of players who choose their efforts ‘early’ and a group of players who choose ‘late’. Only the player with the lowest cost of effort has a positive payoff in any equilibrium. This payoff depends on his own timing vis-a-vis the timing of others. We also show that the choice of timing can be endogenized, in which case the strongest player typically chooses ‘late’, whereas all other players are indifferent with respect to their choice of timing. In the most prominent equilibrium the player with the lowest cost of effort wins the auction at zero aggregate cost. We thank Dan Kovenock and Luis C. Corchón for discussion and helpful comments. The usual caveat applies. Wolfgang Leininger likes to express his gratitude to Wissenschaftszentrum Berlin (WZB) for its generous hospitality and financial support.     

Piecewise closed-loop equilibria in differential games with regime switching strategies

We propose a new methodology exploring piecewise closed-loop equilibrium strategies in differential games with regime switching actions. We develop a general game with two players. Players choose an action that influences the evolution of a state variable, and decide on the switching time from one regime to another. Compared to the optimal control problem with regime switching, necessary optimality conditions are modified for the first player to switch. When choosing her optimal switching strategy, this player considers the impact of her choice on the other player’s actions and consequently on her own payoffs. In order to determine the equilibrium timing of regime changes, we derive conditions that help eliminate candidate equilibrium strategies that do not survive deviations in switching strategies. We then apply this new methodology to an exhaustible resource extraction game.     

The effects of transportation system improvements on urban performances with heterogeneous residents

Through developing a spatial equilibrium model for a linear monocentric city with a bi-modal traffic corridor (i.e., highway and public transit), this paper examines the effects of transportation improvements (including the decreases of the fixed travel cost, the travel time and monetary cost per unit distance) on urban spatial structure and the utilities of the two resident classes, namely car-available-residents (CARs) and car-unavailable-residents (CURs). As a result, it is demonstrated that the city size increases with the improvements of vehicle-highway system and the degree of CARs’ travel mode choice rationality, but the improvement of public transit may produce a shrinkage rather than an expansion in city size. From the aspect of utility, both CARs and CURs benefit from the improvement of vehicle-highway system; however, the improvement of public transit might have an adverse effect on the utility of CARs. These findings would provide valuable reference for the future transportation and urban planning.     

Sad-Loser contests

We consider asymmetric winner-reimbursed contests. It turns out that such contests (Sad-Loser) have multiple internal pure-strategy equilibria (where at least two players are active). We describe all equilibria and discuss their properties. In particular, we find (1) that an active player is indifferent among all her non-negative choices and her expected payoff is zero in any internal equilibrium, (2) that a higher-value (stronger) player always spends less than a lower-value (weaker) player and therefore always has a lower chance to win a Sad-Loser contest in any internal equilibrium, and (3) a sufficient condition for a net total spending to be higher in a Sad-Loser contest than in the corresponding asymmetric contest.     

Deadlines in stochastic contests

We consider a two-player contest model in which breakthroughs arrive according to privately observed Poisson processes. Each player’s process continues as long as she exerts costly effort. The player who collects the most breakthroughs until a predetermined deadline wins a prize.We derive Nash equilibria of the game depending on the deadline. For short deadlines, there is a unique equilibrium in which players use identical cutoff strategies, i.e., they continue until they have a certain number of successes. If the deadline is long enough, the symmetric equilibrium distribution of an all-pay auction is an equilibrium distribution over successes in the contest. Expected efforts may be maximal for a short or intermediate deadline.     

Correlated equilibria in homogeneous good Bertrand competition

We show that there is a unique correlated equilibrium, identical to the unique Nash equilibrium, in the classic Bertrand oligopoly model with homogeneous goods and identical marginal costs. This provides a theoretical underpinning for the so-called “Bertrand paradox” as well as its most general formulation to date. Our proof generalizes to asymmetric marginal costs and arbitrarily many players in the following way: The market price cannot be higher than the second lowest marginal cost in any correlated equilibrium.     

Buying cooperation in an asymmetric environmental differential game

We consider a two-player asymmetric differential game of pollution control. One player is non-vulnerable to pollution, or unwilling to consider damages when choosing her production policy in a non-cooperative game. We characterize the feedback-Nash equilibrium and the cooperative solution. We establish conditions under which the vulnerable player can buy the cooperation of the non-vulnerable player to control her emissions. We further use the Nash bargaining solution to allocate the total cooperative dividend between the two players and propose a time-consistent decomposition overtime of the total payoff.     

Paying For Joint Costs in Health Care

The paper analyzes a regulatory game between a public and a private payer to finance hospital joint costs (mainly capital and technology expenses). The public payer (inspired by the federal Medicare program) may both directly reimburse for joint costs ("pass-through" payments) and add a margin over variable costs paid per discharge, while the private payer can only use a margin policy. The hospital chooses joint costs in response to payers' overall payment incentives. Without pass-through payments, under provision of joint costs results front free-riding behavior of payers and the first-mover advantage of the public payer. Using pass-through policy in its self-interest, the public payer actually may moderate the under provision of joint costs; under some conditions, the equilibrium allocation may be socially efficient. Our results bear directly on directly Medicare policy, which is phasing out pass-through payments.     

An evolutionary model with best response and imitative rules

We formulate an evolutionary oligopoly model where quantity setting players produce following either the static expectation best response or a performance-proportional imitation rule. The choice on how to behave is driven by an evolutionary selection mechanism according to which the rule that brought the highest performance attracts more followers. The model has a stationary state that represents a heterogeneous population where rational and imitative rules coexist and where players produce at the Cournot–Nash level. We find that the intensity of choice, a parameter representing the evolutionary propensity to switch to the most profitable rule, the cost of the best response implementation as well as the number of players have ambiguous roles in determining the stability property of the Cournot–Nash equilibrium. This marks important differences with most of the results from evolutionary models and oligopoly competitions. Such differences should be referred to the particular imitative behavior we consider in the present modeling setup. Moreover, the global analysis of the model reveals that the above-mentioned parameters introduce further elements of complexity, conditioning the convergence toward an inner attractor. In particular, even when the Cournot–Nash equilibrium loses its stability, outputs of players little differ from the Cournot–Nash level and most of the dynamics is due to wide variations of imitators’ relative fraction. This describes dynamic scenarios where shares of players produce more or less at the same level alternating their decision mechanisms.     

Inefficiency and complementarity in sharing games

A sharing game is a very simple device for partially reconciling an organization’s goal with the interests of its members. Each member chooses an action, bears its cost, and receives a share of the revenue which the members’ actions generate. A (pure-strategy) equilibrium of the game may be inefficient: surplus (revenue minus the sum of costs) may be less than maximal. In a previous paper, we found that for a wide class of reward functions, no one squanders at an inefficient equilibrium (spends more than at an efficient profile) if the revenue function has a complementarity property. In the present paper, we examine the “opposite” of the complementarity property (Substitutes) and we study a class of finite games where squandering equilibria indeed occur if Substitutes holds strongly enough. Squandering equilibria play a key role when one traces the effect of technological improvement on a sharing game’s surplus shortfall. We then turn to the question of choice among reward functions in a principal/agents setting. We find that if we again assume complementarity then strong conclusions can be reached about the reward functions preferred by “society”, by the players (agents), and by the principal.     

A psychometric analysis of residential location

Traditional housing needs studies focus on the trade-off between location and housing amenities. In most models, mode choice is viewed as conditional behavior in a given setting. New movers, however, select a new environment which includes a house, neighbourhood and transportation system. By using psychometric techniques, the attitudes and preferences of new movers to the suburbs of a large northeast city were analyzed to determine the relative importance in their selection process of the transportation characteristics of the new location. It was found that local and regional transportation and public transit played little role in selecting an apartment. There was no evidence of tradeoffs between travel time and living space postulated by urban economics. Most important to the choice process of these residents were internal characteristics of the apartment and pricing issues. This supports the idea that suburbanites chose to be captive auto users even when equivalent housing opportunities with transport alternatives are available. Implications for public transit and land use planning alternatives for the suburbs are discussed.     

Competition in Regulated Markets with Sluggish Beliefs about Quality

We investigate the effect of competition on quality in regulated markets (e.g., health care, higher education, public utilities), using a Hotelling framework, in the presence of sluggish beliefs about quality. We take a differential‐game approach, and derive the open‐loop solution (providers choose the optimal quality investment plan based on demand at the initial period) and the feedback closed‐loop solution (providers observe demand in each period and choose quality in response to current demand). If variable costs are strictly convex, and the degree of cost complementarity between quality and output is not too strong, the steady‐state quality is higher under the open‐loop solution than under the feedback solution. In both solutions, quality and demand move in opposite directions over time on the equilibrium path to the steady‐state. While lower transportation costs or less sluggish beliefs lead to higher quality in both solutions, the quality response is weaker when players use feedback strategies.     

Stable cost sharing in production allocation games

Suppose that a group of agents have demands for some good. Every agent owns a technology which allows them to produce the good, with these technologies varying in their effectiveness. If all technologies exhibit increasing returns to scale (IRS) then it is always efficient to centralize production of the good, whereas efficiency in the case of decreasing returns to scale (DRS) typically requires to spread production. We search for stable cost allocations while differentiating allocations with homogeneous prices, in which all units produced are traded at the same price, from allocations with heterogeneous prices. For the respective cases of IRS or DRS, it is shown that there always exist stable cost sharing rules with homogeneous prices. Finally, in the general framework (under which there may exist no stable allocation at all) we provide a sufficient condition for the existence of stable allocations with homogeneous prices. This condition is shown to be both necessary and sufficient in problems with unitary demands.     

Mixed Duopoly with Subcontracting

Existing literature on mixed oligopoly focuses on competition among different types of firms but ignores their possible cooperation. We allow cooperation between a public firm and a private firm through subcontracting in a Hotelling mixed‐duopoly model. We find that when subcontracting is possible, the equilibrium without subcontracting is not socially optimal because subcontracting can lower total production costs. And if both firms engage in subcontracting, the existence of a public firm can guarantee the first best equilibrium, whether it is the low‐cost firm or not. But when a private firm is the low‐cost firm, it is more profitable for it to choose vertical foreclosure. And the consequent equilibrium is not socially desirable anymore. Copyright © 2014 John Wiley & Sons, Ltd.     

Risk preference heterogeneity in group contests

We analyze the first model of a group contest with players that are heterogeneous in their risk preferences. In our model, individuals’ preferences are represented by a utility function exhibiting a generalized form of constant absolute risk aversion, allowing us to consider any combination of risk-averse, risk-neutral, and risk-loving players. We begin by proving equilibrium existence and uniqueness under both linear and convex investment costs. Then, we explore how the sorting of a compatible set of players by their risk attitudes into competing groups affects aggregate investment. With linear costs, a balanced sorting (i.e., minimizing the variance in risk attitudes across groups) always produces an aggregate investment level that is at least as high as an unbalanced sorting (i.e., maximizing the variance in risk attitudes across groups). Under convex costs, however, identifying which sorting is optimal is more nuanced and depends on preference and cost parameters.     

Revealed preferences of individual players in sequential games

This paper studies rational choice behavior of a player in sequential games of perfect and complete information without an assumption that the other players who join the same games are rational. The model of individually rational choice is defined through a decomposition of the behavioral norm assumed in the subgame perfect equilibria, and we propose a set of axioms on collective choice behavior that characterize the individual rationality obtained as such. As the choice of subgame perfect equilibrium paths is a special case where all players involved in the choice environment are each individually rational, the paper offers testable characterizations of both individual rationality and collective rationality in sequential games.