Existence Of Subgame Perfect Equilibrium Research Articles

Competition among Pairwise Lottery Contests

We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a fully polynomial-time approximation scheme to compute an approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.

Multi-lateral strategic bargaining without stationarity

This paper establishes existence of subgame perfect equilibrium in pure strategies for a general class of sequential multi-lateral bargaining games, without assuming a stationary setting. The only required hypothesis is that utility functions are continuous on the space of economic outcomes. In particular, no assumption on the space of feasible payoffs is needed. The result covers arbitrary and even time-varying bargaining protocols (acceptance rules), externalities, and other-regarding preferences. As a side result, we clarify the meaning of assumptions on “continuity at infinity.”

Subgame‐perfect equilibrium in games with almost perfect information: Dispensing with public randomization

Harris, Reny, and Robson (1995) added a public randomization device to dynamic games with almost perfect information to ensure existence of subgame perfect equilibria (SPE). We show that when Nature's moves are atomless in the original game, public randomization does not enlarge the set of SPE payoffs: any SPE obtained using public randomization can be “decorrelated” to produce a payoff‐equivalent SPE of the original game. As a corollary, we provide an alternative route to a result of He and Sun (2020) on existence of SPE without public randomization, which in turn yields equilibrium existence for stochastic games with weakly continuous state transitions.

On spatial competition with quadratic transport costs and one online firm

Guo and Lai (Ann Reg Sci 52(1):309–324, 2014) argue that, under quadratic transport costs, when two offline firms compete with one online firm, the two offline firms locate in such a way that they occupy unconnected regions in the market. However, we offer a counterexample to show that their provided condition is not sufficient to support existence of subgame perfect equilibria, because location deviations to different market structures are not taken into account.

Subgame Perfection and the Rule of k Names

In this paper we revisit the rule of k names from a game theoretic perspective. This rule can be described as follows. Given a set of candidates for a position, a committee (formed by the proposers) selects k elements of that set using a screening rule; then a single individual from outside the committee (the chooser) chooses for the position one of the k selected candidates. In this context we first give conditions for the existence of a subgame perfect equilibrium. Then we provide conditions for the existence of subgame perfect q-strong equilibria when the screening rule is $$\pi $$ -majoritarian. Finally, we show that when the chooser can strategically appoint a delegate to choose on behalf of him, the conditions for the existence of subgame perfect q-strong equilibria are weaker.

Subgame-Perfect Equilibrium in Games with Almost Perfect Information: Dispensing with Public Randomization

Harris, Reny, and Robson (Econometrica, 1995) add a public randomization device to dynamic games with almost perfect information to ensure existence of subgame perfect equilibria (SPE). We show that when nature's moves are atomless in the original game, SPE obtained via this channel can be de-correlated to produce a payoff-equivalent SPE of the original game; thus, public randomization is in a sense without loss of generality. As a corollary, we provide an alternative route to a result of He and Sun (2016) on existence of SPE without public randomization, which in turn yields an equilibrium existence result for stochastic games with weakly continuous state transitions.

Stationary Almost Markov Perfect Equilibria in Discounted Stochastic Games

In this paper, we study discounted stochastic games with Borel state and compact action spaces depending on the state variable. The primitives of our model satisfy standard continuity and measurability conditions. The transition probability is a convex combination of finitely many probability measures depending on states, and it is dominated by some finite measure on the state space. The coefficients of the combination depend on both states and action profiles. This class of models contains stochastic games with Borel state spaces and finite, state-dependent action sets. Our main result establishes the existence of subgame perfect equilibria, which are stationary in the sense that the equilibrium strategy for each player is determined by a single function of the current and previous states of the game. This dependence is called almost Markov. Our result enhances both the theorem of Mertens and Parthasarathy established in 1991 for games with finite, state-independent action sets, where the equilibrium strategies were also depended on the calendar time, and their result on stationary equilibria proved under an additional condition that the transition probabilities are atomless. A counterexample given very recently by Levy shows that stationary Markov perfect equilibria may not exist in the class of games considered in this paper. The presented results in this work are illustrated by the Cournot dynamic games, which were already considered in the literature under much stronger assumptions.

Backward induction and unacceptable offers

How to establish the existence of subgame perfect equilibrium (SPE) in bargaining models if no stationary SPEs (SSPEs) exist? The backward-induction technique of Shaked and Sutton (1984, Econometrica) applies to the cyclical structure of SPE payoffs and provides recursive dynamics on the bounds of SPE payoffs. Acceptable and unacceptable offers have to be incorporated for these dynamics to be necessary and sufficient for extreme SPEs. In this paper, we demonstrate how these recursive dynamics are directly applicable to establish the existence of SPE in a model with no SSPE. Also from these dynamics, the extreme SPE strategy profiles can easily be recovered.

A game theoretic analysis of the cops and robber game

We provide a game theoretic framework for the game of cops and robbers. Within this framework we study certain assumptions which underlie the concepts of optimal strategies and capture time. We show rigorously that these assumptions are justified. Our results are based on the theory developed by C. Alos-Ferrer and K. Ritzberger on the existence of subgame perfect equilibria in extensive form games which are infinite and / or have infinite horizon.

On (Subgame Perfect) Secure Equilibrium in Quantitative Reachability Games

We study turn-based quantitative multiplayer non zero-sum games played on finite graphs with reachability objectives. In such games, each player aims at reaching his own goal set of states as soon as possible. A previous work on this model showed that Nash equilibria (resp. secure equilibria) are guaranteed to exist in the multiplayer (resp. two-player) case. The existence of secure equilibria in the multiplayer case remained and is still an open problem. In this paper, we focus our study on the concept of subgame perfect equilibrium, a refinement of Nash equilibrium well-suited in the framework of games played on graphs. We also introduce the new concept of subgame perfect secure equilibrium. We prove the existence of subgame perfect equilibria (resp. subgame perfect secure equilibria) in multiplayer (resp. two-player) quantitative reachability games. Moreover, we provide an algorithm deciding the existence of secure equilibria in the multiplayer case.

Overlapping Coalitions, Bargaining and Networks

This paper extends the theory of endogenous coalition formation, with complete information and transferable utility, to the overlapping case. We propose a cover function bargaining game which allows the formation of overlapping coalitions at equilibrium. We show the existence of subgame perfect equilibrium and provide an algorithm to compute this equilibrium in the symmetric case. As an application, we establish an interesting link with the formation of networks.

Existence of equilibrium in common agency games with adverse selection

We establish the existence of subgame perfect equilibria in general menu games, known to be sufficient to analyze common agency problems. Our main result states that every menu game satisfying enough continuity properties has a subgame perfect equilibrium. Despite the continuity assumptions that we make, discontinuities naturally arise due to the absence, in general, of continuous optimal choices for the agent. Our approach, then, is based on (and generalizes) the existence theorem of [Simon, L., Zame, W., 1990. Discontinuous games and endogenous sharing rules. Econometrica 58 (4), 861–872] designed for discontinuous games.

A Simple Proof of a Theorem by Harris

We present a simple proof of existence of subgame perfect equilibria in games with perfect information.

BARGAINING MODEL WITH SEQUENCES OF DISCOUNT RATES AND BARGAINING COSTS

The paper is a kind of generalization of Rubinstein bargaining model. Rubinstein assumed that preferences of the players were constant in time, and he analyzed models in which preferences of each player were defined either by constant discount rate or by constant bargaining cost. In this paper, a bargaining model is presented, in which preferences of each player are expressed simultaneously by sequence of discount rates and sequence of bargaining costs varying in time. The results presented in the paper concern subgame perfect equilibria. There is a theorem concerning sufficient and necessary conditions for the existence of subgame perfect equilibrium of the game. Moreover, some theorems presenting forms of subgame perfect equilibria for various cases of the model analyzed have been proved here. A possibility of delay in reaching an agreement is also considered in the paper. If we analyze a class of strategies, that depend on the former history, a delay can appear for some models. The adequate examples are presented. In the paper, some applications of the bargaining model are also described.

The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Comment

HARRIS (1985) HAS SHOWN that subgame-perfect equilibria exist in deterministic continuous games with perfect information.' A recent influential paper by Harris, Reny, and Robson (1995) shows that public randomization ensures the existence of subgameperfect equilibria in continuous games with almost perfect information. The authors exhibit an example of a game with almost perfect information in which no subgameperfect equilibrium exists without public randomization. In addition, their Proposition 36 argues that public randomization is not required in games with perfect information. Contrary to this proposition, we give here an example of a continuous game with perfect information in which no subgame-perfect equilibrium exists if public randomization is not allowed. The result of Harris (1985) implies that our example must be a game in which Nature is an active player. Intrapersonal games for consumers with changing preferences are usually games of this type, as in, for example, stochastic versions of Peleg and Yaari (1973) and Goldman (1980). The example has five stages. In stage 1, player 1 chooses al E [0, 1]. In stage 2, player 2 chooses a2 E [0, 1]. In stage 3, Nature chooses x by randomizing uniformly over the interval [-2 + a, + a2, 2 - a, - a2]. After this, players 3 and 4 move sequentially. The subgame following a history (a,, a2, x) and the associated payoffs for all four players are shown in Figure 1. In the following, let a and ,B denote the probabilities with which players 3 and 4 choose U and u, respectively. Consider first the subgame defined by a, = a2 = 1. Nature's move is degenerate in this subgame: x = 0. The set of subgame-perfect equilibrium paths in the resulting subgame is characterized by three segments of mixing probabilities: a = 0 and /3 E [0, .5]; a E [0, 1] and ,B = .5; and a = 1 and f3 E [.5, 1]. The set of equilibrium expected payoffs for players 1 and 2 implied by these three segments is given by

The effect of punishment duration of trigger strategies and quasifinite continuation probabilities for Prisoners' Dilemmas

Limiting inspection to publicly correlated strategies linearizes the problem of finding subgame perfect equilibria for indefinitely repeated Prisoners' Dilemmas. Equilibrium strategies satisfy a matrix inequality; the matrix depends on the duration of the punishment mechanism. The existence of subgame perfect equilibria is determined by comparing a parameter of the stage game to the spectral radius of the matrix. The spectral radii are found for the matrices associated with every possible duration of punishment.

Subgame Perfect Equilibria and Communication in Stage Games

Any stage-game with infinite choice sets can be approximated by finite games obtained as increasingly finer discretizations of the infinite game. The subgame perfect equilibrium outcomes of the finite games converge to a limit distribution. We prove that (i) if the limit distribution is feasible in the limit game, then it is also a subgame perfect equilibrium outcome of the limit game; and (ii) if the limit distribution prescribes sufficiently diffused behavior for first-stage players, then it is a subgame perfect equilibrium outcome of the limit game. These results are potentially useful in determining the existence of subgame perfect equilibria in applications. As an illustration of this potential, it is shown that the addition of cheap talk to the games considered restores the existence of subgame perfect equilibria.

The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Case for Public Randomization

The starting point of this paper is a simple, regular dynamic game in which subgame-perfect equilibrium fails to exist. Examination of this example shows that existence would be restored if players were allowed to observe the output of a public-randomization device. The main result of the paper shows that the introduction of public randomization yields existence not only in the example, but also in a large class of dynamic games. It is also argued that the introduction of public randomization is the minimal robust extension of subgame-perfect equilibrium in this class of games.

Symmetric equilibrium existence and optimality in differentiated product markets

Equilibrium existence and optimality are analysed in a market for products differentiated by their variety. A game of three stages is analysed. Firms enter in the first stage, choose varieties in the second stage, and choose prices in the third stage. The existence of subgame-perfect equilibria is established. At equilibrium products are symmetrically located in the space of characteristics and are offered at equal prices. The surplus maximizing solution is characterized, and it is shown that surplus maximizing product diversity is lower than the equilibrium one.