Pure strategy equilibria in a class of systems defense games

Abstract

This paper analyzes a class of systems defense games. Section 1 provides an overview. Section 2 reviews the assumptions used to specify a class of system defense games in Shubik and Weber (1978, 1981) and Shubik (1982). This is followed by a review of the additive games considered in Shubik and Weber (1981) -and the notion of comparable amounts of strategic resources used in that paper. Section 3 generalizes two of the features of additive system defense games with comparable resources. Section 4 states the final assumptions that are being made, and discusses the nature of the resulting class of systems defense games. The central result in the paper is in Section 5. The theorem specifically identifies an implicit objective function that is maximized by the equilibrium strategies. In Section 6, the theorem is used to also obtain an equilibrium existence result and identify conditions where there is a unique equilibrium.

Conflicts with Multiple Battlefields

“…Alternating move models of sunk expenditure within a single contest environment in the spirit of the Shubik "Dollar Auction Game" (see O'Neill (1986) and Leininger (1989Leininger ( , 1991) have been extended to cover multiple battlefield conflicts (in the context of vote buying) by Dekel et al (2008Dekel et al ( , 2009). 21 Finally, the fact that the main results of this chapter provide equilibrium payoffs in terms of exogenous parameters such as resource budgets, number of battlefields, values of battlefields, unit costs of expenditures, and network structures, facilitates the use of these models as the final stage of multistage games in which these values are determined endogenously. This opens up a conflict theoretic framework with which to generate equilibrium models of the endogenous determination of the configuration of battlefields (including endogenous network structures and network redundancies), 22 resources constraints and cost structures, 23 and coalition formation.…”

“…A line of research that examines games of attack and defense with multiple battlefields in which the players have asymmetric objective functions includes Gross (1950), Cooper and Restrepo (1967), Shubik and Weber (1981), Coughlin (1992), and Clark and Konrad (2007). 15 For example, Clark and Konrad (2007) use the terms weakest-link objective and best-shot objective to describe the objectives of a defender and an attacker, respectively, who face each other in a network of battlefields in which successful defense requires that all targets within the network be successfully defended and successful attack requires only that at least one battlefield be won.…”

Sequential, nonzero-sum “Blotto”: Allocating defensive resources prior to attack

“…The more a player dedicates to a particular front, the higher the probability of prevailing on that front (e.g., Shubik and Weber, 1981). Blotto games with continuous payoff functions may have purestrategy equilibria (Blackett, 1958;Shubik and Weber, 1978;Coughlin, 1992). Indeed, the defender's unique equilibrium strategy in the static version of the game studied here is pure although the attacker does mix (Powell, 2007).…”

Choosing What to Protect: Strategic Defensive Allocation against an Unknown Attacker

“…10 10 Major's model belongs to a class traditionally called Colonel Blotto games : zero‐sum games involving two players with fixed resources and n battlefields (see Shubik and Weber 1981 for a classic reference, and Coughlin 1992 for a generalization). …”