Games of Threats

Elon   Kohlberg; Abraham  Neyman

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Article | Games and Economic Behavior

Games of Threats

by Elon Kohlberg and Abraham Neyman

Abstract

A game of threats on a finite set of players, N, is a function d that assigns a real number to any coalition, S ⊆ N, such that d(S) = -d(N\S). A game of threats is not necessarily a coalitional game as it may fail to satisfy the condition d(Ø) = 0. We show that analogs of the classic Shapley axioms for coalitional games determine a unique value for games of threats. This value assigns to each player an average of the threat powers, d(S), of the coalitions that include the player. Games of threats arise naturally in value theory for strategic games and may have applications in other branches of game theory.

Keywords: Shapley value; coalitional game; Game Theory; Mathematical Methods;

Format: Print

10 pages

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Citation:

Kohlberg, Elon, and Abraham Neyman. "Games of Threats." Games and Economic Behavior 108 (March 2018): 139–145.

About the Author

Elon Kohlberg

Royal Little Professor of Business Administration, Emeritus

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Working Paper | HBS Working Paper Series | 2020

Demystifying the Math of the Coronavirus

Elon Kohlberg and Abraham Neyman

We provide an elementary mathematical description of the spread of the coronavirus. We explain two fundamental relationships: How the rate of growth in new infections is determined by the “effective reproductive number” and how the effective reproductive number is affected by social distancing. By making a key approximation, we are able to formulate these relationships very simply and thereby avoid complicated mathematics. The same approximation leads to an elementary method for estimating the effective reproductive number.

Keywords: coronavirus; Health Pandemics; Mathematical Methods;

Citation:
Kohlberg, Elon, and Abraham Neyman. "Demystifying the Math of the Coronavirus." Harvard Business School Working Paper, No. 20-112, April 2020. (Revised May 2020.) View Details

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Working Paper | HBS Working Paper Series | 2017

Cooperative Strategic Games

Elon Kohlberg and Abraham Neyman

We examine a solution concept, called the “value," for n-person strategic games. In applications, the value provides an a-priori assessment of the monetary worth of a player's position in a strategic game, comprising not only the player's contribution to the total payoff but also the player's ability to inflict losses on other players. A salient feature is that the value takes account of the costs that “spoilers" impose on themselves. Our main result is an axiomatic characterization of the value. For every subset, S, consider the zero-sum game played between S and its complement, where the players in each of these sets collaborate as a single player, and where the payoff is the difference between the sum of the payoffs to the players in S and the sum of payoffs to the players not in S. We say that S has an effective threat if the minmax value of this game is positive. The first axiom is that if no subset of players has an effective threat then all players are allocated the same amount. The second axiom is that if the overall payoff to the players in a game is the sum of their payoffs in two unrelated games, then the overall value is the sum of the values in these two games. The remaining axioms are the strategic-game analogs of the classical coalitional-games axioms for the Shapley value: efficiency, symmetry, and null player.

Keywords: Game Theory;

Citation:
Kohlberg, Elon, and Abraham Neyman. "Cooperative Strategic Games." Harvard Business School Working Paper, No. 17-075, February 2017. View Details

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Working Paper | HBS Working Paper Series | 2015

The Cooperative Solution of Stochastic Games

Elon Kohlberg and Abraham Neyman

Building on the work of Nash, Harsanyi, and Shapley, we define a cooperative solution for strategic games that takes account of both the competitive and the cooperative aspects of such games. We prove existence in the general (NTU) case and uniqueness in the TU case. Our main result is an extension of the definition and the existence and uniqueness theorems to stochastic games—discounted or undiscounted.

Keywords: game theory; economics; Game Theory;

Citation:
Kohlberg, Elon, and Abraham Neyman. "The Cooperative Solution of Stochastic Games." Harvard Business School Working Paper, No. 15-071, March 2015. View Details

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Games of Threats

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