In their engrossing article, De Dreu and Gross might have overlooked some directly relevant literature on asymmetric games of conflict. In particular, they largely disregard the literature on (generalised) matching pennies (MP) games (only citing Goeree et al. Reference Goeree, Holt and Palfrey2003 in passing). In section 2.1, especially Figure 1, they appear to define the binary Attacker-Defender Game (AD-G) as a combination of the assurance game and the game of Chicken; this leads them to keep identical labels of cooperation and defection for both players’ strategies in AD-G, which is debatable given that the game is asymmetric. Much more importantly, assurance and Chicken games may not be the nearest relatives of the binary AD-G that have been considered in the literature; this AD-G is just an example of the well-known class of generalised MP games (also referred to as inspection games or police-public games). In either case, one of the players wants to match and the other wants to mismatch (see Table 1) so that the only equilibrium is one in mixed strategies, which is, however, Pareto inefficient. The main difference would be in the framing (which may matter in such games; see Eliaz & Rubinstein Reference Eliaz and Rubinstein2011). Indeed, the labels of “attacker”/”defender” are not typically used in (generalised) MP games and, in some of these games, the strategies could not reasonably be labelled “cooperation” and “defection,” because when the “defender” is “protected,” it may actually benefit from the “attacker” “seeking gain”/”attacking.”
Table 1. De Dreu and Gross's AD-G (A) and Goeree et al.’s Game 4 (scaled down and with rows reversed to facilitate a comparison), a generalised MP game (B)
Arguably, each of the three possibilities (the defender being indifferent when protected, as in Table 1A; the defender being hurt by the attack even when protected, although not as much as when unprotected, as in Table 1B; the defender benefitting from the attack when protected, as in standard MP games) may be the most appropriate, depending on the conflict situation at hand. For example, the cost of war will usually still be serious for the defender, even if it is well prepared and ultimately victorious; on the other hand, such an episode may strengthen its international position as a powerful player and scare other potential aggressors away, and so on. Similar consideration applies to the attacker's preferences concerning the defender's actions when the attacker does not seek gain.
A number of findings from MP games parallel those reported by D&G. In particular, players tend to react in natural ways to their opponent's probability of choosing each option in the past (e.g., Colman Reference Colman1999). Moreover, in Goeree et al.’s Game 4, shown here in Table 1B, “attackers” (column players) choose the “defective” strategy Right (hurting the other player) significantly less often than the Nash equilibrium would require, but there is no analogous effect for the “defenders.” This is equivalent to D&G's findings in continuous AD-Gs. Then again, the evidence for “attackers” being relatively timid is rather mixed in Dorris and Glimcher (Reference Dorris and Glimcher2004), Rauhut (Reference Rauhut2009), and Nosenzo et al. (Reference Nosenzo, Offerman, Sefton and van der Veen2013).
D&G's claim that “theory and research has rarely made a clear distinction between attack and defense” (sect. 1, para. 4) is also potentially slightly misleading when being applied to models of conflict with continuous action space. Indeed, several papers have analysed asymmetric conflict situations both theoretically (Franke et al. Reference Franke, Kanzow, Leininger and Schwartz2013; Nti Reference Nti1999) and empirically (Carter & Anderton Reference Carter and Anderton2001; Dechenaux et al. Reference Dechenaux, Kovenock and Sheremeta2015, p. 623). One frequently considered type of heterogeneity is that of a player possessing more resources and trying to keep them whilst another seeks their redistribution, which may be naturally interpreted as a defence-attack situation. Further, in “multi-battle” contexts, the goals of the defender and the attacker are often explicitly differentiated (Deck & Sheremeta Reference Deck and Sheremeta2012; Kovenock et al. Reference Kovenock, Roberson and Sheremeta2010).
Finally, there is literature looking at process data (such as reaction times, eye-tracking, and neuroimaging) in MPs (Hampton et al. Reference Hampton, Bossaerts and O'Doherty2008; Krol & Krol Reference Krol and Krol2017). While these typically involve the standard zero-sum MP game, still the player who tries to match may be thought of as a defender, while the one who tries to mismatch may naturally be construed as an attacker. For example, D&G's proposition that defenders act more spontaneously than attackers is preceded by the data of Martin et al. (Reference Martin, Bhui, Bossaerts, Matsuzawa and Camerer2014), in whose experiments the matchers were faster than the mismatchers. These authors interpret it in terms of humans’ automatic tendency to imitate (Belot et al. Reference Belot, Crawford and Heyes2013). To recapitulate, a more complete survey of the relevant extant literature, even if it involves different labels in analogous games, could provide insights into the robustness of D&G's reported findings and their interpretations.
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In their engrossing article, De Dreu and Gross might have overlooked some directly relevant literature on asymmetric games of conflict. In particular, they largely disregard the literature on (generalised) matching pennies (MP) games (only citing Goeree et al. Reference Goeree, Holt and Palfrey2003 in passing). In section 2.1, especially Figure 1, they appear to define the binary Attacker-Defender Game (AD-G) as a combination of the assurance game and the game of Chicken; this leads them to keep identical labels of cooperation and defection for both players’ strategies in AD-G, which is debatable given that the game is asymmetric. Much more importantly, assurance and Chicken games may not be the nearest relatives of the binary AD-G that have been considered in the literature; this AD-G is just an example of the well-known class of generalised MP games (also referred to as inspection games or police-public games). In either case, one of the players wants to match and the other wants to mismatch (see Table 1) so that the only equilibrium is one in mixed strategies, which is, however, Pareto inefficient. The main difference would be in the framing (which may matter in such games; see Eliaz & Rubinstein Reference Eliaz and Rubinstein2011). Indeed, the labels of “attacker”/”defender” are not typically used in (generalised) MP games and, in some of these games, the strategies could not reasonably be labelled “cooperation” and “defection,” because when the “defender” is “protected,” it may actually benefit from the “attacker” “seeking gain”/”attacking.”
Table 1. De Dreu and Gross's AD-G (A) and Goeree et al.’s Game 4 (scaled down and with rows reversed to facilitate a comparison), a generalised MP game (B)
Arguably, each of the three possibilities (the defender being indifferent when protected, as in Table 1A; the defender being hurt by the attack even when protected, although not as much as when unprotected, as in Table 1B; the defender benefitting from the attack when protected, as in standard MP games) may be the most appropriate, depending on the conflict situation at hand. For example, the cost of war will usually still be serious for the defender, even if it is well prepared and ultimately victorious; on the other hand, such an episode may strengthen its international position as a powerful player and scare other potential aggressors away, and so on. Similar consideration applies to the attacker's preferences concerning the defender's actions when the attacker does not seek gain.
A number of findings from MP games parallel those reported by D&G. In particular, players tend to react in natural ways to their opponent's probability of choosing each option in the past (e.g., Colman Reference Colman1999). Moreover, in Goeree et al.’s Game 4, shown here in Table 1B, “attackers” (column players) choose the “defective” strategy Right (hurting the other player) significantly less often than the Nash equilibrium would require, but there is no analogous effect for the “defenders.” This is equivalent to D&G's findings in continuous AD-Gs. Then again, the evidence for “attackers” being relatively timid is rather mixed in Dorris and Glimcher (Reference Dorris and Glimcher2004), Rauhut (Reference Rauhut2009), and Nosenzo et al. (Reference Nosenzo, Offerman, Sefton and van der Veen2013).
D&G's claim that “theory and research has rarely made a clear distinction between attack and defense” (sect. 1, para. 4) is also potentially slightly misleading when being applied to models of conflict with continuous action space. Indeed, several papers have analysed asymmetric conflict situations both theoretically (Franke et al. Reference Franke, Kanzow, Leininger and Schwartz2013; Nti Reference Nti1999) and empirically (Carter & Anderton Reference Carter and Anderton2001; Dechenaux et al. Reference Dechenaux, Kovenock and Sheremeta2015, p. 623). One frequently considered type of heterogeneity is that of a player possessing more resources and trying to keep them whilst another seeks their redistribution, which may be naturally interpreted as a defence-attack situation. Further, in “multi-battle” contexts, the goals of the defender and the attacker are often explicitly differentiated (Deck & Sheremeta Reference Deck and Sheremeta2012; Kovenock et al. Reference Kovenock, Roberson and Sheremeta2010).
Finally, there is literature looking at process data (such as reaction times, eye-tracking, and neuroimaging) in MPs (Hampton et al. Reference Hampton, Bossaerts and O'Doherty2008; Krol & Krol Reference Krol and Krol2017). While these typically involve the standard zero-sum MP game, still the player who tries to match may be thought of as a defender, while the one who tries to mismatch may naturally be construed as an attacker. For example, D&G's proposition that defenders act more spontaneously than attackers is preceded by the data of Martin et al. (Reference Martin, Bhui, Bossaerts, Matsuzawa and Camerer2014), in whose experiments the matchers were faster than the mismatchers. These authors interpret it in terms of humans’ automatic tendency to imitate (Belot et al. Reference Belot, Crawford and Heyes2013). To recapitulate, a more complete survey of the relevant extant literature, even if it involves different labels in analogous games, could provide insights into the robustness of D&G's reported findings and their interpretations.