Equilibria of Attacker-Defender Games

Abstract

Attempts at predatory capture may provoke a defensive response that reduces the very value of the predated resource. We provide a game-theoretic analysis of simultaneous-move, two-player Attacker-Defender games that model such interactions. When initial endowments are equal, Attackers win about a third of such games in equilibrium. Under power disparities, Attackers become particularly aggressive when they are approximately one-third poorer than Defenders. With non-conflictual outside options Attackers become exceptionally aggressive when their opponent has access to high-benefit, low-cost production, and refrain from attack most when they are unilaterally provided with a high-benefit, high-cost production option.

On being unpredictable and winning.

“…Our main variable of interest was cross-round variability in conflict investment. As noted above, the attacker–defender contest has its equilibrium in mixed strategies, and we have precise probability estimates for each investment level x (0 ≤ x ≤ e ) (De Dreu et al, 2015, 2021; Meder et al, 2023). We operationalized cross-round variability in two complementary ways.…”

“…, where f is the probability that the attacker wins, m → ∞ for x A ≠ x D and f = 0 if x D = x A (De Dreu et al, 2015;Meder et al, 2023).…”

“…In contrast to stylized agents assumed to be motivated and capable to rationally maximize expected payoffs, people may be limited in the degree to which they vary actions both with the same or with different opponents. For example, people may want to avoid the risk of losing and therefore, compared to “homo economicus” agents, reduce the amount they invest in attack and increase the amount they invest in defense (Chowdhury et al, 2018; Meder et al, 2023; Sheremeta, 2013; Yang et al, 2020). In both cases, the range of possible actions is reduced, and with that variability in conflict behavior may be reduced as well.…”

“…When x A ≤ x D , both attacker and defender earn their noninvested resources ( e − x A , x D ). As such, the game is formally equivalent to a contest with as contest success function f = x A m /( x A m + x D m ), where f is the probability that the attacker wins, m → ∞ for x A ≠ x D and f = 0 if x D = x A (De Dreu et al, 2015; Meder et al, 2023).…”

“…Specifically, the probability p of investing x in attack is p ( x = 1) = 2/45, p ( x ) = p ( x − 1)[(12 − x )/(10 − x )] for 2 ≤ x ≤ 6, and p ( x ) = 0 for x ≥ 7. For defense, the probability p of investing y in defense is p ( y ) = 1/(10 − y ) for 0 ≤ y ≤ 5, p ( y = 6) = 1 − [ p ( y = 0) + … + p ( y = 5)] = 0.15, and p ( y ) = 0 for y ≥ 7 (De Dreu et al, 2015, 2021; Meder et al, 2023). This implies that participants should invest, on average, x A = 2.62 in attack and x D = 3.38 in defense and that across trials, investments should vary with variance v A = 6.12 for attack and v D = 3.68 for defense.…”