Leader-follower Equilibrium Research Articles

Endogenous Price Leadership with Asymmetric Costs: Experimental Evidence

This paper presents experimental evidence on the action commitment game with cost-asymmetric firms in a differentiated-products Bertrand duopoly. Unlike its quantity-setting counterpart, the risk-dominant leader–follower equilibrium Pareto dominates the simultaneous-move equilibrium. This equilibrium also minimizes payoff differences between firms. Hence, one would expect the model to accurately capture behavior. The evidence partially supports the theory: low-cost firms price in the first period more often than high-cost firms, and depending on the treatment, between 40 and 57 per cent of all observations conform to equilibrium play. However, the modal timing outcome involved both firms delaying their pricing decision. This timing outcome is characterized by Nash play and some collusion. The high frequency of delaying decisions could be due to a desire to reduce strategic uncertainty.

Computing the Strategy to Commit to in Polymatrix Games

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games—polymatrix like—which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies.

Leaders, Followers, and Risk Dynamics in Industry Equilibrium

AbstractWe study the distinct impacts of own and rival actions on risk and return when firms strategically compete in the product market. Contrary to simple intuition, a competitor’s options to adjust capacity reduce own-firm risk. For example, if a rival possesses a growth option, an increase in industry demand directly enhances profits but also encourages value-reducing competitor expansion. The rival option thus acts as a natural hedge. Within the industry, we obtain endogenous differences in expected returns. In a leader-follower equilibrium, own-firm and competitor risks and required returns move together through contractions and oppositely during expansions, providing testable new predictions.

Relative difference contest success function

In this paper we present a Contest Sucess Function (CSF) which is homogeneous of degree zero and where the probabilities of winning the prize depend on the relative difference of efforts. With two agents, we present a necessary and sufficient condition for the existence of a Nash Equilibrium in pure strategies. This equilibrium is unique and interior. This condition does not depend on the size of the valuations as in absolute difference CSF. We prove that several properties of Nash equilibrium and Leader-Follower equilibrium with the Tullock CSF still hold in our framework. Finally, we consider the case of n players, generalize the previous condition and show that it is sufficient for the existence of a unique Nash equilibrium in pure strategies.

Relative Difference Contest Success Function

In this paper we present a Contest Sucess Function (CSF) which is homogeneous of degree zero and where the probabilities of winning the prize depend on the relative difference of efforts. With two agents, we present a necessary and sufficient condition for the existence of a Nash Equilibrium in pure strategies. This equilibrium is unique and interior. This condition does not depend on the size of the valuations as in absolute difference CSF. We prove that several properties of Nash equilibrium and Leader-Follower equilibrium with the Tullock CSF still hold in our framework. Finally, we consider the case of n players, generalize the previous condition and show that it is sufficient for the existence of a unique Nash equilibrium in pure strategies.

Patrolling security games: Definition and algorithms for solving large instances with single patroller and single intruder

Security games are gaining significant interest in artificial intelligence. They are characterized by two players (a defender and an attacker) and by a set of targets the defender tries to protect from the attackerʼs intrusions by committing to a strategy. To reach their goals, players use resources such as patrollers and intruders. Security games are Stackelberg games where the appropriate solution concept is the leader–follower equilibrium. Current algorithms for solving these games are applicable when the underlying game is in normal form (i.e., each player has a single decision node). In this paper, we define and study security games with an extensive-form infinite-horizon underlying game, where decision nodes are potentially infinite. We introduce a novel scenario where the attacker can undertake actions during the execution of the defenderʼs strategy. We call this new game class patrolling security games (PSGs), since its most prominent application is patrolling environments against intruders. We show that PSGs cannot be reduced to security games studied so far and we highlight their generality in tackling adversarial patrolling on arbitrary graphs. We then design algorithms to solve large instances with single patroller and single intruder.

Leaders, Followers, and Risk Dynamics in Industry Equilibrium

We study the distinct impacts of own and rival actions on risk and return when firms strategically compete in the product market. Contrary to simple intuition, a competitor’s options to adjust capacity reduce own-firm risk. For example, if a rival possesses a growth option, an increase in industry demand directly enhances profits but also encourages value-reducing competitor expansion. The rival option thus acts as a natural hedge. Within the industry, we obtain endogenous differences in expected returns. In a leader-follower equilibrium, own-firm and competitor risks and required returns move together through contractions and oppositely during expansions, providing testable new predictions.

Unilateral Spillovers between East and West and Quality Competition

We model a two-stage duopolistic competition in a vertically differentiated ekstern market between the eastern and western firms. In the first stage firms compete in RD the eastern firm increases the quality of its product solely through imitation. We show that under assumptions (1).(3) listed above and for absorption rate close to 1, the welfare level in the eastern country is higher when the eastern firm imitates only compared to the welfare level in the leader-follower equilibrium.

Price Commitment versus Flexibility: the Role of Exchange Rate Uncertainty and its Implications for Exchange Rate Passthrough

AbstractThe paper investigates the incentives to commit price or retain price flexibility in a model in which exporting firms face different degrees of exchange rate uncertainty. The result shows that introducing exchange rate uncertainty can lead to the endogenous emergence of a unique leader–follower equilibrium; which firm emerges as price leader depends on the substitutability of products, the magnitude of exchange rate uncertainty, and the cost structure. This study may provide one explanation as to why some exporters set price before the realization of the nominal exchange rates (“sticky price”). The results imply exchange rate variability affects exchange rate passthrough.

Strategic Delay in a Real Options Model of R&D Competition

This paper considers irreversible investment in competing research projects with uncertain returns under a winner-takes-all patent system. Uncertainty takes two distinct forms: the technological success of the project is probabilistic, while the economic value of the patent to be won evolves stochastically over time. According to the theory of real options uncertainty generates an option value of delay, but with two competing firms the fear of preemption would appear to undermine this approach. In non-cooperative equilibrium two patterns of investment emerge depending on parameter values. In a pre-emptive leader-follower equilibrium firms invest sequentially and option values are reduced by competition. A symmetric outcome may also occur, however, in which investment is more delayed than the single-firm counterpart. Comparing this with the optimal cooperative investment pattern, investment is found to be more delayed when firms act non-cooperatively as each holds back from investing in the fear of starting a patent race. Implications of the analysis for empirical and policy issues in R&D are considered.

The Timing and Terms of Takeovers Under Uncertainty: A Real Options Approach

This paper presents a dynamic model for the timing and terms of mergers, stock offers and cash offers when the output price for firms is stochastic and when firms have complete information. We show that takeovers motivated by increased returns to scale or by decreased unit production costs are procyclical. Mergers are modelled as the outcome of a Nash equilibrium in which both companies simultaneously determine the timing and terms of the restructuring, whereas stock and cash offers are the outcome of a Stackelberg leader follower equilibrium in which the target first fixes its reservation bid premium and the acquirer subsequently decides on the timing of the takeover. For mergers, each firm's cumulative return consists of a synergy effect that rises with the increase in its production scale parameter. With tender offers, the target's synergy effect is augmented by a bid premium that is increasing in the output price volatility, the acquirer's return and the bidder to target size ratio. The timing of mergers is shown to be globally efficient. Separation of the timing decision and the terms decision causes stock offers and cash offers to happen inefficiently late. Pure cash offers happen even later than stock offers because the acquirer's payoff has optionlike features, whereas the target's has not. Delaying the restructuring therefore favors the acquirer at the expense of the target.

Navigating a nonlinear environment: An experimental study of decision making in a chaotic setting

This paper presents experimental results where subjects are placed in a nonlinear game setting with the potential for chaotic dynamics in best-responses. We explore what behavior emerges from the nonlinear environment, how players make choices in this complex environment, and what strategies are most successful. We find an assortment of dynamic play in players' choices, ranging from an emergent leader–follower equilibrium to `chaotic' dynamics. We also find that players who most successfully navigated this complex decision environment, achieving the highest payoffs, were those who relied on an incremental search strategy.

Price leadership and asymmetric price rigidity

In this paper we present a simple price leadership model in which equilibrium behavior exhibits price rigidity following downward demand shocks and price flexibility after an upturn in demand. The source of this asymmetric rigidity lies in the fact that leader-follower equilibrium prices are lower than their collusive levels and that any firm leading a round of price adjustment must anticipate the optimal price response of the follower. In addition, we find that there is a range of shocks, both positive and negative, in which the identity of the price leader is endogenous; the previous price leader is the only firm that prefers leading a round of price setting to keeping the status quo.

A leader-follower model and analysis for a two-stage network of oligopolies

This paper is concerned with the existence, uniqueness and computation of leader-follower equilibrium solutions for an industry involved with two major stages of production. We assume that there exists one set of firms performing the first stage of production, which produces a semi-finished product. This semi-finished product is converted to a final good by a second set of firms performing the second stage of production. Furthermore, also competing in the final product market is a third set of firms, which are vertically integrated through the two stages of production and which are assumed to lead the second set of firms by explicitly considering the reaction or response of these latter firms to their own outputs. We model such an industry as a two-stage network of oligopolies, and define equilibrium solutions based on assumed market structures. Our analysis examines the existence and uniqueness of such equilibrium solutions, characterizes the nature of the production strategies of the various firms at an equilibrium, and prescribes algorithms to compute such solutions. This provides the machinery required to perform sensitivity analyses for studying the effects of various mergers or integrations on individual firm profits, and on the industry outputs and prices at equilibrium. The presentation is self-contained, and does not necessarily require any significant prior preparation in economic theory on the part of the reader.