Inter and intra-group conflicts as a foundation for contest success functions

This paper introduces a notion of partitioned correlated equilibrium that extends Aumann's correlated equilibrium concept (1974, 1987). This concept captures the non-cooperative interactions arising simultaneously within and between groups. We build on this notion in order to provide a foundation for contest success functions (CSFs) in a game wherein contests arise endogenously. Our solution concept and analysis are general enough to give a foundation for any model of contest using standard equilibrium concepts like e.g., Nash, Bayesian-Nash or Perfect-Nash equilibria. In our environment, popular CSFs can be interpreted as a list of equilibrium conjectures held by players whenever they contemplate deviating from the ``peaceful outcome'' of the ``group formation game''. Our setup allows to relate the form of prominent CSFs with some textbook examples of quasi-linear utility functions, social utility functions in the spirit of Fehr and Schmidt (1999) and non-expected models of utility a la Quiggin (1981, 1982). We also show that our framework can accommodate situations in which agents cannot correlate their actions.Contest success functions; Correlated equilibrium; Inter and intra-group conflicts; Induced contests

Mediated Contests and Strategic Foundations for Contest Success Functions

This paper examines the foundations of arbitrary contest success functions (CSFs) in two distinct types of contests – unmediated and mediated contests. In an unmediated contest, CSFs arise as the (interim) players’ equilibrium beliefs of a two-stage game – the gun-butter game – in which players choose an activity (appropriative vs. productive) in the first stage, and apply effort to their activity in the second stage. In this view a CSF is rationalizable if a contest is induced on the equilibrium path of the gun-butter game. In the second approach, a CSF is the result of the optimal design of an administrator. Here, the designer seeks to maximize his utility by implementing a probability distribution on the set of contestants. However, he is curbed by a disutility term which represents the underlying institutional constraints or the designer’s preferences. Both approaches provide foundations for arbitrary CSFs with no restrictions on the number of contestants.Induced contests; Gun-butter game; Control costs

CORRELATED EQUILIBRIUM STRATEGIES WITH MULTIPLE INDEPENDENT RANDOMIZATION DEVICES

A primitive assumption underlying Aumann (1974,1987) is that all players of a game may correlate their strategies by agreeing on a common single ’public roulette’. A natural extension of this idea is the study of strategies when the assumption of a single random device common to all the players (public roulette) is dropped and (arbitrary) disjoint subsets of players forming a coalition structure are allowed to use independent random devices (private roulette) a la Aumann. Under multiple independent random devices, the coalition's mixed strategies form an equilibrium of the induced non-cooperative game played across the coalitions–the ’partitioned game’–when the profile of such coalitions’ strategies is a profile of correlated equilibria. These correlated equilibria which are the mutual joint best responses of the coalitions are called the Nash coalitional correlated equilibria (NCCEs) of the game. The paper identifies various classes of finite and infinite games where there exists a non-empty set of NCCEs lying outside the regular correlated equilibrium distributions of the game. We notably relate the class of NCCEs to the ’coalitional equilibria’ introduced in Ray and Vohra (1997) to construct their ’Equilibrium Binding Agreements’. In a ’coalitional equilibrium’, coalitions’ best responses are defined by Pareto dominance and their existence are not guaranteed in arbitrary games without the use of correlated mixed strategies. We characterize a family of games where the existence of a non-empty set of non-trivial NCCEs is guaranteed to coincide with a subset of coalitional equilibria. Most of our results are based on the characterization of the induced non-cooperative ’partitioned game’ played across the coalitions

Ontological foundation of Nash Equilibrium

In the classical definition of a game, the players' hierarchies of beliefs are not part of the description. So, how can a player determine a rational choice if beliefs are initially nonexistent in his mind? We address this question in a three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. This ``rationalistic'' Kripke structure permits to study the ``mental states'' of players when they consider the perspectives or decision problems of the others, in order to form their own beliefs. In our main Theorem we provide necessary and sufficient conditions for Nash equilibrium in an n-person game. This proves that the initial indeterminism of the game model, together with the free will of rational players are at the origin of this concept. This equivalence result has several implications. First, this demonstrates that a Nash equilibrium is not an interactive solution concept but an intrinsic principle of decision making used by each player to shape his/her own beliefs. Second, this shows that a rational choice must be viewed in statu nascendi i.e. conceived as a genuine ``act of creation'' ex nihilo, rather than as a pre-determined decision, arising from an underlying history of the game

Inter and intra-group conflicts as a foundation for contest success functions

This paper introduces a notion of partitioned correlated equilibrium that extends Aumann's correlated equilibrium concept (1974, 1987). This concept captures the non-cooperative interactions arising simultaneously within and between groups. We build on this notion in order to provide a foundation for contest success functions (CSFs) in a game wherein contests arise endogenously. Our solution concept and analysis are general enough to give a foundation for any model of contest using standard equilibrium concepts like e.g., Nash, Bayesian-Nash or Perfect-Nash equilibria. In our environment, popular CSFs can be interpreted as a list of equilibrium conjectures held by players whenever they contemplate deviating from the ``peaceful outcome'' of the ``group formation game''. Our setup allows to relate the form of prominent CSFs with some textbook examples of quasi-linear utility functions, social utility functions in the spirit of Fehr and Schmidt (1999) and non-expected models of utility a la Quiggin (1981, 1982). We also show that our framework can accommodate situations in which agents cannot correlate their actions

Mediated Contests and Strategic Foundations for Contest Success Functions

This paper examines the foundations of arbitrary contest success functions (CSFs) in two distinct types of contests – unmediated and mediated contests. In an unmediated contest, CSFs arise as the (interim) players’ equilibrium beliefs of a two-stage game – the gun-butter game – in which players choose an activity (appropriative vs. productive) in the first stage, and apply effort to their activity in the second stage. In this view a CSF is rationalizable if a contest is induced on the equilibrium path of the gun-butter game. In the second approach, a CSF is the result of the optimal design of an administrator. Here, the designer seeks to maximize his utility by implementing a probability distribution on the set of contestants. However, he is curbed by a disutility term which represents the underlying institutional constraints or the designer’s preferences. Both approaches provide foundations for arbitrary CSFs with no restrictions on the number of contestants

Inter and intra-group conflicts as a foundation for contest success functions

This paper introduces a notion of partitioned correlated equilibrium that extends Aumann's correlated equilibrium concept (1974, 1987). This concept captures the non-cooperative interactions arising simultaneously within and between groups. We build on this notion in order to provide a foundation for contest success functions (CSFs) in a game wherein contests arise endogenously. Our solution concept and analysis are general enough to give a foundation for any model of contest using standard equilibrium concepts like e.g., Nash, Bayesian-Nash or Perfect-Nash equilibria. In our environment, popular CSFs can be interpreted as a list of equilibrium conjectures held by players whenever they contemplate deviating from the ``peaceful outcome'' of the ``group formation game''. Our setup allows to relate the form of prominent CSFs with some textbook examples of quasi-linear utility functions, social utility functions in the spirit of Fehr and Schmidt (1999) and non-expected models of utility a la Quiggin (1981, 1982). We also show that our framework can accommodate situations in which agents cannot correlate their actions

Mediated Contests and Strategic Foundations for Contest Success Functions

This paper examines the foundations of arbitrary contest success functions (CSFs) in two distinct types of contests – unmediated and mediated contests. In an unmediated contest, CSFs arise as the (interim) players’ equilibrium beliefs of a two-stage game – the gun-butter game – in which players choose an activity (appropriative vs. productive) in the first stage, and apply effort to their activity in the second stage. In this view a CSF is rationalizable if a contest is induced on the equilibrium path of the gun-butter game. In the second approach, a CSF is the result of the optimal design of an administrator. Here, the designer seeks to maximize his utility by implementing a probability distribution on the set of contestants. However, he is curbed by a disutility term which represents the underlying institutional constraints or the designer’s preferences. Both approaches provide foundations for arbitrary CSFs with no restrictions on the number of contestants

Ontological foundation of Nash Equilibrium

In the classical definition of a game, the players' hierarchies of beliefs are not part of the description. So, how can a player determine a rational choice if beliefs are initially nonexistent in his mind? We address this question in a three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. This ``rationalistic'' Kripke structure permits to study the ``mental states'' of players when they consider the perspectives or decision problems of the others, in order to form their own beliefs. In our main Theorem we provide necessary and sufficient conditions for Nash equilibrium in an n-person game. This proves that the initial indeterminism of the game model, together with the free will of rational players are at the origin of this concept. This equivalence result has several implications. First, this demonstrates that a Nash equilibrium is not an interactive solution concept but an intrinsic principle of decision making used by each player to shape his/her own beliefs. Second, this shows that a rational choice must be viewed in statu nascendi i.e. conceived as a genuine ``act of creation'' ex nihilo, rather than as a pre-determined decision, arising from an underlying history of the game