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Equilibrium and efficiency in the tug-of-war

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  • Konrad, Kai Andreas
  • Kovenock, Daniel J.

Abstract

We characterize the unique Markov perfect equilibrium of a tug-of-war without exogenous noise, in which players have the opportunity to engage in a sequence of battles in an attempt to win the war. Each battle is an all-pay auction in which the player expending the greater resources wins. In equilibrium, contest effort concentrates on at most two adjacent states of the game, the tipping states, which are determined by the contestants' relative strengths, their distances to final victory, and the discount factor. In these states battle outcomes are stochastic due to endogenous randomization. Both relative strength and closeness to victory increase the probability of winning the battle at hand. Patience reduces the role of distance in determining outcomes. Applications range from politics, economics and sports, to biology, where the equilibrium behavior finds empirical support: many species have developed mechanisms such as hierarchies or other organizational structures by which the allocation of prizes are governed by possibly repeated conflict. Our results contribute to an explanation why. Compared to a single stage conflict, such structures can reduce the overall resources that are dissipated among the group of players. -- Wir beschreiben das eindeutige Markoff-perfekte Gleichgewicht in einem mehrstufigen Konflikt ohne exogene Unsicherheit (noise), bei dem die Spieler versuchen, in einer Serie von aufeinander folgenden kleineren Gefechten einen Konflikt zu gewinnen. Jedes Gefecht ist eine all-pay auction, bei der derjenige Spieler gewinnt, der die meisten Ressourcen eingesetzt hat. Im Gleichgewicht konzentriert sich der Mitteleinsatz auf höchstens zwei benachbarte Zustände, die wir als spielentscheidende Zustände (tipping states) bezeichnen. Die Lage dieser Zustände hängt von der relativen Stärke der Spieler, der Zahl der verbleibenden Spielstufen bis zum Gesamtsieg und dem Diskontierungsfaktor ab. An diesen kritischen Zuständen ist der Konfliktausgang zufällig aufgrund der stochastischen Verteilung der im Gleichgewicht gewählten Mengen von Konfliktressourcen. Sowohl die relative Stärke als auch die Nähe zur finalen Konfliktstufe erhöhen die Wahrscheinlichkeit, das einzelne Gefecht zu gewinnen. Geringe Kosten des Wartens verringern den Einfluss der Entfernung zum Gesamtsieg auf den Ausgang der einzelnen Gefechte. Die Anwendungsgebiete sind zahlreich und reichen von der Politik über die Wirtschaft und den Sport bis zur Biologie. Dort findet das Gleichgewichtsergebnis empirisch Unterstützung: Viele Arten haben eigene Mechanismen entwickelt, z.B. Hierarchien oder andere Organisationsstrukturen, bei denen die Allokation der Siegerprämie in sich möglicherweise wiederholenden Konflikten erfolgt. Unsere Ergebnisse liefern hierzu eine Erklärung. Im Vergleich mit einem einstufigen Konflikt können solche Strukturen den Ressourceneinsatz der Spieler reduzieren.

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Bibliographic Info

Paper provided by Social Science Research Center Berlin (WZB) in its series Discussion Papers, Research Unit: Market Processes and Governance with number SP II 2005-14.

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Date of creation: 2005
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Handle: RePEc:zbw:wzbmpg:spii200514

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Keywords: Winner-take-all; all-pay auction; tipping; multi-stage contest; dynamic game; pre-emption; conflict; dominance;

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