Formalizing Anger

Anger is thought to affect both preferences and information processing (Mintz, Valentino, and Wayne Reference Mintz, Valentino and Wayne2021). We focus here on the effect of anger on preferences, though we consider the effects of anger on information processing in section “Informational Effects of Anger.”

Anger’s role in shaping preferences is linked to the perceptions that trigger it. Cognitive appraisal theorists have found that individuals are most likely to experience anger when they deem an event to be negative (i.e., against their interests) and purposefully caused by another (Lazarus Reference Lazarus1991; Roseman Reference Roseman1996; Scherer Reference Scherer, Dalgleish and Power1999). Anger is also a moral emotion, elicited not simply when a negative event occurs, but when that event is seen as unfair (Haidt Reference Haidt, Davidson, Scherer and Hill Goldsmith2003). As such, anger is related to assessments of the other’s character or intentions: individuals are more likely to experience anger when they perceive the actions of another to be unjust or their motivations illegitimate.Footnote ⁴

A challenge in formally modeling anger-triggering perceptions is establishing the benchmark for negative events. We assume, consistent with previous work, that individuals gauge an event as negative if it falls short of the expected outcome (Scherer Reference Scherer, Hamilton, Bower and Frijda1988). The frustration-aggression hypothesis posits that barriers to achieving an expected gratification lead to aggression (Berkowitz and Harmon-Jones Reference Berkowitz and Harmon-Jones2004; Dollard et al. Reference Dollard, Miller, Doob, Mowrer and Sears1939). Battigalli and Dufwenberg (Reference Battigalli and Dufwenberg2009) concur, emphasizing the role of expectations in evaluating outcomes. Empirical data underpin this context-dependent perception of anger: greater discrepancy between expectations and reality boosts the likelihood of anger (Benistant and Suchon Reference Benistant and Suchon2021; Persson Reference Persson2018). This phenomenon is visible in the escalation of violence following unanticipated losses in various contexts (Barnhart Reference Barnhart2020; Card and Dahl Reference Card and Dahl2011) and explains the frequency of interpersonal anger toward liked and respected others, where unmet high expectations foster disappointment and anger (Averill Reference Averill1983).

Once triggered, anger fosters a desire to punish perceived wrongdoers, driven more by psychological satisfaction than material benefits (Carlsmith, Darley, and Robinson Reference Carlsmith, Darley and Robinson2002; Lerner and Keltner Reference Lerner and Keltner2001; McDermott, Lopez, and Hatemi Reference McDermott, Lopez and Hatemi2017). This is because anger promotes deontological thinking, where actions adhere to moral rules rather than consequences (García-Ponce, Young, and Zeitzoff Reference García-Ponce, Young and Zeitzoff2023). Angry individuals may prioritize “nonmaterial stakes” and be willing to forgo material gains to ensure wrongdoers face repercussions (Hall Reference Hall2021). This behavior is evident in ultimatum bargaining games where unfair yet materially beneficial offers are often rejected, a tendency that is reduced when anger is mitigated (Pillutla and Keith Murnighan Reference Pillutla and Keith Murnighan1996; Srivastava, Espinoza, and Fedorikhin Reference Srivastava, Espinoza and Fedorikhin2009). Fairness heuristics, which are central to anger, can also influence decisions to escalate diplomatic disputes to war (Gottfried and Trager Reference Gottfried and Trager2016). Broadly, anger alters actor preferences, leading to dissatisfaction with benefits received by the adversary and potentially engendering cycles of retaliatory political violence (Gollwitzer et al. Reference Gollwitzer, Skitka, Wisneski, Sjöström, Liberman, Nazir and Bushman2014; Liberman and Skitka Reference Liberman and Skitka2017; Pagano and Huo Reference Pagano and Huo2007).

However, to fully understand how anger shapes conflict decision-making, we should place anger in a strategic context: how does anger shape the dynamic interaction between strategic actors in conflict? This is the question we address in this paper.

Behavioral Game Theory in Politics

Our work also contributes to the growing literature incorporating behavioral assumptions into formal models of political phenomena (e.g., Acharya, Blackwell, and Sen Reference Acharya, Blackwell and Sen2018; Bendor, Kumar, and Siegel Reference Bendor, Kumar and Siegel2010; Feddersen, Gailmard, and Sandroni Reference Feddersen, Gailmard and Sandroni2009; Little Reference Little2019; Minozzi Reference Minozzi2013; Penn Reference Penn2008; Siegel Reference Siegel2011) and research on context-dependent preferences (e.g., Agranov et al. Reference Agranov, Goeree, Romero and Yariv2017; Alesina and Passarelli Reference Alesina and Passarelli2019; Kahneman and Tversky Reference Kahneman and Tversky1979).

Methodologically, our paper is most related to those that endogenize preferences through some psychological mechanism. The main examples in international relations are Acharya and Grillo (Reference Acharya and Grillo2019) and Haynes and Yoder (Reference Haynes and Yoder2022). Acharya and Grillo (Reference Acharya and Grillo2019) considers endogenous “disappointment” as a foundation for audience costs, a phenomenon where politicians are punished for making a threat and then backing down. As with anger in this paper, disappointment is triggered when an outcome is not in line with an agent’s expectations. Though our paper is methodologically related to this work, we focus on a different application and on the agents actively engaged in conflict. Haynes and Yoder (Reference Haynes and Yoder2022) focus on a strategic setting very similar to ours but the modeling of endogenous preferences differs in two important ways. First, preferences in our model are endogenous in the sense of depending on strategies in a way that is incorporated into the solution concept using psychological game theory. In contrast, Haynes and Yoder (Reference Haynes and Yoder2022) let preferences change only as a function of past actions, in a manner much more similar to the alternative model we explain in section “Comparison to Naive Anger Preferences.” Second, the model in Haynes and Yoder (Reference Haynes and Yoder2022) allows for positive and negative preference adaptations which are not important for the purpose of our paper.Footnote ⁷

Finally, our work also relates to models of reciprocity using psychological games (Dufwenberg and Kirchsteiger Reference Dufwenberg and Kirchsteiger2004; Rabin Reference Rabin1993). Agents motivated by reciprocity prefer to be kind to agents who they perceive as kind and unkind to agents they perceive as unkind. To model anger, we focus on the negative side of reciprocity, that is, the desire to be unkind in response to perceived unkindness. Anger has been related to preferences for negative reciprocity, particularly across identity groups (Bicskei, Lankau, and Bizer Reference Bicskei, Lankau and Bizer2016), so our perspective is in line with this research.

Sequence and Game Play

Consider a game between overlapping generations of agents from two groups, A and B, interacting over an infinite time horizon with time indexed by $ t\in \{0,1,\dots \hskip0.3em \}. $ In each period t, one player is active (“player t”) and, for $ t>0 $ , the active player chooses an action $ {x}_t\in \{c,a\} $ (“conciliatory” or “aggressive”) toward player $ t-1 $ and an intended action $ {y}_t\in \{c,a\} $ toward player $ t+1 $ . Player 0 only chooses an intended action $ {y}_0 $ toward player 1. In odd periods, the active player is from group A; and in even periods, the active player is from group B, so all actions are directed toward members of the opposite group.

In any period $ t>0 $ , player t learns a result $ {\tilde{y}}_{t-1}. $ The observed result is stochastic but depends on the intended action in that $ \Pr [{\tilde{y}}_{t-1}=a|{y}_{t-1}=a]=1 $ and $ \Pr [{\tilde{y}}_{t-1}=a|{y}_{t-1}=c]=\pi $ . In short, aggressive intentions will lead to aggressive results, but conciliatory intentions may, with some probability ( $ \pi $ ), mistakenly also lead to aggressive results. Player 0 gets no signal and simply chooses an intended action.

Player t learns her own type and the result $ {\tilde{y}}_{t-1} $ but nothing else about the history of the game. Agents do not learn about any interactions that came before and also do not know how much time passed before she became active (i.e., agents do not know the value of t).Footnote ⁸ This assumption incorporates the poor information and high degree of uncertainty about adversaries’ present and past actions that often exists in conflict contexts (Ramsay Reference Ramsay2017), but this assumption is relaxed in section “More Information about History.” Following Acemoglu and Wolitzky (Reference Acemoglu and Wolitzky2014), we formalize ignorance of calendar time by assuming that each player at $ t>0 $ has an improper uniform prior over time periods, which implies that the prior probability of observing a particular value of $ {\tilde{y}}_{t-1} $ is equal to the long-run probability of that outcome.

Types, Material Payoffs, and Strategies

Player t has a type $ {\theta}_t\in \{H,F\} $ (“Hostile” or “Friendly”). Hostile types have a dominant strategy of aggression. In terms of material payoffs, Friendly types prefer to match the action of the other player. The proportion of Hostile types in group $ j\in \{A,B\} $ is $ {\rho}_j $ . Players’ types are private information. Additionally, players are uncertain about the proportion of Hostile types in each group, denoted $ {\rho}_A $ and $ {\rho}_B $ . We assume for each $ j\in \{A,B\} $ that $ {\rho}_j $ takes a value of either $ \underset{\_}{\rho } $ or $ \overline{\rho}>\underset{\_}{\rho }. $ For both groups, the prior probability that $ {\rho}_j=\overline{\rho} $ is $ {\mu}_0. $ We typically assume that $ \overline{\rho}=1 $ and $ \underset{\_}{\rho }=0 $ , so that any group is made of either all Hostile types or all Friendly types. This is in line with Acemoglu and Wolitzky (Reference Acemoglu and Wolitzky2014) and leads to the clearest exposition of results, though the proofs of Propositions 1 and 2 are for general values of $ \overline{\rho} $ and $ \underset{\_}{\rho } $ .Footnote ⁹

Each player $ t>0 $ gets a material payoff of $ {v}_t=u({x}_t,{\tilde{y}}_{t-1},{\theta}_t)+u({y}_t,{x}_{t+1},{\theta}_t), $ where u describes the payoff from each of the two interactions in which that player participates. We set $ u(a,a)=0 $ and $ u(c,c)=1 $ , representing the idea that all players, including Hostile types, would materially prefer the situation in which both players are conciliatory to the situation in which both players are aggressive. Friendly players prefer coordination. If the other player is aggressive, the payoff to acting conciliatory is $ -s, $ where $ s>0 $ represents a cost of unilaterally taking a conciliatory action. Finally, the payoff to acting aggressive when the other player is conciliatory is $ r({\theta}_t) $ , where $ r(F)<1<r(H). $ Thus, Friendly types prefer to reciprocate conciliatory behavior but Hostile types always prefer aggression. Players’ payoffs depend on their own chosen actions and the result of the actions (but not the intended action) of the other players. That is, player t’s payoffs are a function of $ {\tilde{y}}_{t-1} $ rather than $ {y}_{t-1} $ : players are materially affected by aggressive outcomes even if they were the result of mistakes. Figure 1 summarizes the material payoffs to a given player.

Figure 1.

Material Payoffs of Player t from Interactions with Players $ t-1 $ and $ t+1 $

Note: Parameter values are $ s>0 $ and $ r(F)<1<r(H). $

We analyze equilibria in which all members of the same group use the same type-dependent strategy. A strategy $ {\sigma}_j $ for group j maps types $ {\theta}_t $ and results $ {\tilde{y}}_{t-1} $ into probabilities of playing $ {x}_t=a $ and $ {y}_t=a. $ Each strategy is an element of the set $ \Sigma ={[0,1]}^8 $ since it prescribes probabilities of two different actions each across four different information sets. The strategy for group A is denoted $ {\sigma}_A $ , the strategy for group B is denoted $ {\sigma}_B $ , and a strategy profile is simply $ \sigma =({\sigma}_A,{\sigma}_B)\in {\Sigma}^2. $

Psychological Motivations

We augment the model described above with psychological motivations to understand how anger affects the onset and continuation of conflict. As we have discussed, empirical research in political psychology has found that anger is most likely when agents observe a negative difference between expected and actual outcomes that they attribute to the actions of another person (Lazarus Reference Lazarus1991; Roseman Reference Roseman1996; Scherer Reference Scherer, Dalgleish and Power1999). Anger also shapes agents’ preferences by increasing their psychological payoff when the adversary’s material payoff decreases (Carlsmith, Darley, and Robinson Reference Carlsmith, Darley and Robinson2002). Anger therefore links preferences with conjectures about players’ strategies, which leads us toward psychological game theory. Our approach follows Battigalli, Dufwenberg, and Smith (Reference Battigalli, Dufwenberg and Smith2019) with modifications to account for incomplete information.

We define a psychological game by directly modeling players’ conjectures about others’ strategies.Footnote ¹⁰ Let $ {b}_t^j\in \Sigma $ be player t’s conjecture about group j’s strategy. That is, for some pair of strategies $ {\sigma}_A^{\prime}\in \Sigma $ and $ {\sigma}_B^{\prime}\in \Sigma $ , $ {b}_t^A={\sigma}_A^{\prime } $ and $ {b}_t^B={\sigma}_B^{\prime } $ means that player’ t thinks that members of group A use the strategy $ {\sigma}_A^{\prime } $ and members of group B use the strategy $ {\sigma}_B^{\prime }. $

In our psychological game, a player may become angry when $ {\tilde{y}}_{t-1}=a $ . The extent of that anger depends on the payoff she expected to receive and how much she blames player $ t-1 $ for this undesirable outcome (Battigalli, Dufwenberg, and Smith Reference Battigalli, Dufwenberg and Smith2019). Since player t’s best possible material payoff from the interaction with $ t-1 $ is $ 0 $ when $ {\tilde{y}}_{t-1}=a, $ anger is proportional to the ex ante expected payoff from the interaction with $ t-1 $ multiplied by the probability that player $ t-1 $ chose $ {y}_{t-1}=a. $ We therefore define the anger level $ \beta $ of player t from group j as shown in Equation 1.

(1) $$ \begin{array}{rl}{\beta}_t({\tilde{y}}_{t-1},{y}_{t-1};{b}_t^j,{b}_t^{-j}):=\left\{\begin{array}{ll}\Pr [{\tilde{y}}_{t-1}=c|{b}_t^j,{b}_t^{-j}]\Pr [{y}_{t-1}=a|{\tilde{y}}_{t-1}=a,{b}_t^j,{b}_t^{-j}],\hskip1em \qquad & \hskip0.3em \mathrm{if}\hskip0.3em {\tilde{y}}_{t-1}=a,\qquad \\ {}0,\hskip1em \qquad & \hskip0.3em \mathrm{if}\hskip0.3em {\tilde{y}}_{t-1}=c,\qquad \end{array}\right.& \end{array} $$

That is, a player’s anger is zero if they see a good outcome ( $ {\tilde{y}}_{t-1}=c $ ). If they see a bad outcome ( $ {\tilde{y}}_{t-1}=a $ ), two factors determine their anger level. The first factor is the deviation between the agent’s expected outcome and the observed outcome. Since the best payoff when $ {\tilde{y}}_{t-1}=c $ is 1 and the best payoff when $ {\tilde{y}}_{t-1}=a $ is zero, the deviation from expectations after getting a bad signal is equal to the prior probability that $ {\tilde{y}}_{t-1}=c $ given players’ conjectured strategies. The second factor is whether the negative outcome was caused by the actions of the other player. However, since player t never observes $ {y}_{t-1} $ , our actual evaluation of player t’s anger level in the solution involves calculating $ \Pr [{y}_{t-1}=a|{\tilde{y}}_{t-1}=a,{b}_t^j,{b}_t^{-j}] $ , as in Equation 1.Footnote ¹¹

Endogenous Preferences and Solution Concept

Each player’s decision utility incorporates material payoffs and psychological motivations. The decision utility for player t is

(2) $$ \begin{array}{r}{u}_t({x}_t,{y}_t,{\tilde{y}}_{t-1};{b}_t^j,{b}_t^{-j})={\displaystyle \underset{\mathrm{Expected}\ \mathrm{material}\ \mathrm{payoff}}{\underbrace{\unicode{x1D53C}[{v}_t|{x}_t,{y}_t,{\tilde{y}}_{t-1},{b}_t^j,{b}_t^{-j}]}}}\\ {}-{\displaystyle \underset{\mathrm{Expected}\ \mathrm{psychological}\ \mathrm{payoff}}{\underbrace{\alpha_j\unicode{x1D53C}[{\beta}_t({\tilde{y}}_{t-1},{y}_{t-1};{b}_t^j,{b}_t^{-j})({v}_{t-1}+{v}_{t+1})]}}}& \end{array} $$

for $ j\in \{A,B\}. $ That is, each player maximizes her own expected material payoff minus a psychological payoff that depends on her anger level and the material payoffs of the players with whom she interacts. The parameter $ {\alpha}_j\ge 0 $ measures the weight placed on psychological motives. The players’ preferences are endogenous in the sense that they depend (through $ {\beta}_t $ ) on the players’ conjectures about the strategies being employed in the game.Footnote ¹²

Beliefs $ {\mu}_t $ for each player map realizations of $ {\theta}_t $ and $ {\tilde{y}}_{t-1} $ to beliefs about $ {\rho}_A $ and $ {\rho}_B $ . An equilibrium is a profile of strategies $ \sigma *=({\sigma}_A^{*},{\sigma}_B^{*}) $ , conjectures $ {b}_t^A $ and $ {b}_t^B $ for each player, and beliefs $ {\mu}_t $ for each player which satisfy the following:

1. 1. Both groups’ strategies maximize each player’s expected decision utility given conjectures $ {b}_t^A $ and $ {b}_t^B $ and beliefs $ {\mu}_t $ : $ {\sigma}_j^{*}({\theta}_t,{\tilde{y}}_{t-1})\in {\displaystyle \underset{\sigma \in \Sigma}{\mathrm{arg}\ \mathrm{max}}}{\unicode{x1D53C}}_{\mu_t}[{u}_t({x}_t,{y}_t,{\tilde{y}}_{t-1};{b}_t^j,{b}_t^{-j})] $ for all $ j\in \{A,B\} $ and all t.

2. 2. For every t, $ {\mu}_t $ is consistent with Bayes rule given conjectures $ {b}_t^A $ and $ {b}_t^B $ at any information set on the path of play.

3. 3. Conjectures and strategies are consistent: $ {b}_t^A={\sigma}_A^{*} $ and $ {b}_t^B={\sigma}_B^{*} $ for all $ t. $

If $ {\alpha}_A={\alpha}_B=0, $ then psychological preferences are irrelevant and this definition corresponds to perfect Bayesian equilibrium. Otherwise, the equilibrium concept follows psychological sequential equilibrium which is developed in Battigalli and Dufwenberg (Reference Battigalli and Dufwenberg2009) and Battigalli, Corrao, and Dufwenberg (Reference Battigalli, Corrao and Dufwenberg2019). Table 1 summarizes the mathematical notation and interpretation for each quantity in the model.Footnote ¹³

Table 1.

Notation from the Model

Note: For functions, “values taken” gives the range of the function.

Foundational Results

Our first result lays out the structure of equilibria and mirrors results in Acemoglu and Wolitzky (Reference Acemoglu and Wolitzky2014).Footnote ¹⁴ First, Hostile agents are always aggressive. Friendly types prefer to reciprocate conciliatory and aggressive actions so Friendly types match $ {x}_t $ to the result of the previous action. To choose an action $ {y}_t $ toward player $ t+1 $ , Friendly agents t update their beliefs about the probability player $ t+1 $ is Hostile. If player $ t+1 $ is likely to be Hostile, then conciliatory actions are unlikely to be reciprocated and the agent should choose aggression. If player $ t+1 $ is likely to be Friendly, then the agent should try a conciliatory action in hopes of reciprocation. Therefore, Friendly types are conciliatory when the result of the previous action was conciliatory since they believe Friendly types in the other group are more likely. When the result of the previous action was aggressive, Friendly types may be aggressive. Proposition 1 states the result.

Proposition 1 makes no reference to anger and these results can be explained solely using expectations of material payoffs. This is because anger nudges agents in the same direction as their beliefs: agents are only angry when $ {\tilde{y}}_{t-1}=a $ , so anger strengthens the tendency to respond to aggression with more aggression but does not affect actions taken when $ {\tilde{y}}_{t-1}=c. $

Proposition 1 leaves open the question of whether certain conflict can occur. Suppose Friendly types from both groups are always aggressive when $ {\tilde{y}}_{t-1}=a $ (i.e., $ {p}_A={p}_B=1 $ ). Then the long-run probability of an aggressive outcome in a particular time period is one: we must eventually observe $ {\tilde{y}}_{t-1}=a $ even if by mistake, after which all players choose $ {y}_t=a $ in every period. Proposition 2, however, shows that this cannot happen in equilibrium.

To explain Proposition 2, recall that two mechanisms cause Friendly types to be aggressive toward the next player following $ {\tilde{y}}_{t-1}=a. $ First, an informational mechanism might lead to aggression because Friendly agents update their beliefs that members of the opposite group are Hostile. Second, an emotional mechanism might lead to aggression because an aggressive result causes a Friendly type to prefer punishing the other group. A situation of certain conflict would eliminate both of these mechanisms. The informational mechanism is eliminated because, since players have limited knowledge about the past and the probability of conflict is one regardless of players’ types, observing $ {\tilde{y}}_{t-1}=a $ reveals nothing about whether the opposite group is Hostile. Therefore, since this is the only event the player observes, the player’s beliefs are equal to her prior which, by Assumption 1, implies she chooses conciliation. The emotional mechanism is eliminated because anger requires a gap between observed and expected outcomes. In a situation of certain conflict, $ {\tilde{y}}_{t-1}=a $ is the expected outcome so it does not generate an angry response.

Proposition 2 demonstrates how our approach to modeling anger constrains agents’ behavior. Though anger has the straightforward consequence of making agents more aggressive, the model cannot simply predict any level of conflict in a trivial manner by increasing the influence of anger. This is methodologically useful because it gives the model falsifiable empirical content that also reflects the real-world tendency of conflicts to ebb and flow, pause and recur (Goertz, Jones, and Diehl Reference Goertz, Jones and Diehl2005; Mason et al. Reference Mason, Gurses, Brandt and Quinn2011). As we will later show in Proposition 4, a simpler alternative approach to modeling anger would not have this key feature.

Effects of Sensitivity to Anger

The next results concern the effects of anger. We show that both groups change their behavior when one group is made more sensitive to anger (i.e., when $ {\alpha}_j $ increases for some group j). In applications, we can think of changing $ {\alpha}_j $ in different ways. If trait-based anger varies across individuals (Kassinove et al. Reference Kassinove, Roth, Owens and Ryan Fuller2002; Keys and Yorke Reference Keys and Yorke2019) or social groups (Gault and Sabini Reference Gault and Sabini2000; Matsumoto, Yoo, and Chung Reference Matsumoto, Yoo, Chung, Potegal, Stemmler and Spielberger2010; McDermott Reference McDermott2015; Phoenix Reference Phoenix2019), then we can interpret comparative statics on $ {\alpha}_j $ as a way to compare predictions across settings where decision-makers vary according to these traits. In experiments, changes in $ {\alpha}_j $ may be generated by priming treatments designed to induce subjects to think more about anger (Banks and Valentino Reference Banks and Valentino2012; Wayne Reference Wayne2023; Zeitzoff Reference Zeitzoff2014). In policy-making settings, variation in $ {\alpha}_j $ may be generated by differences in personality of various leaders (George and George Reference George and George2019; Horowitz, Stam, and Ellis Reference Horowitz, Stam and Ellis2015; Saunders Reference Saunders2011) or of institutional structures that emphasize more centralized decision-making by a single leader versus slow deliberation across a bureaucracy (Greenstein Reference Greenstein1967; Hafner-Burton et al. Reference Hafner-Burton, Haggard, Lake and Victor2017; Powell Reference Powell2017) which, if the effects of anger depend on quick unitary action, may generate differences in practice in sensitivity to anger. Thus, a flexible interpretation of the model can lend itself to a variety of applications.

Changing a group’s sensitivity to anger has two effects on the behavior of the players. First, changing one group’s sensitivity to anger directly makes members of that group more aggressive by giving them more punitive preferences when they experience an aggressive outcome. Second, changing one group’s sensitivity to anger has an indirect effect on members of the other group. This effect runs in the opposite direction: increasing one group’s sensitivity to anger makes Friendly members of the other group less aggressive. Proposition 3 states the result.

Why does increasing one group’s sensitivity to anger make the other group less aggressive? It is not because anger necessarily has a deterrent effect (Sell, Tooby, and Cosmides Reference Sell, Tooby and Cosmides2009). Rather, our model highlights a different core mechanism for how one side’s anger can affect the other’s behavior: the information conveyed by aggression from Friendly types. When a player observes an aggressive outcome, they update in favor of the other group being Hostile, and therefore become more aggressive. Now consider an increase in $ {\alpha}_A, $ which makes Friendly group A members more aggressive. This reduces the informational effect of an aggressive outcome so Friendly group B members are then less likely to be swayed into aggression through changing beliefs. The other side could have acted aggressively because they were Hostile; but they also could have acted aggressively because they were simply temporarily angry. Furthermore, since increasing $ {\alpha}_A $ increases the likelihood that a group B member experiences an aggressive outcome, aggressive outcomes induce a less severe anger response, as the gap between expectations and outcomes is smaller. Agents expect that the other side may be more likely to choose aggression. Both mechanisms lead in the direction of less aggression from one group when the other is known to be sensitive to anger.

We offer four additional remarks on the interpretation of Proposition 3. First, the results track the distinction between intrapersonal effects of anger, which refer to the influence of an emotion on the person experiencing it, and interpersonal, which refer to the effects on individuals with whom the person interacts (Morris and Keltner Reference Morris and Keltner2000; Van Kleef et al. Reference Van Kleef, Van Dijk, Steinel, Harinck and Van Beest2008). The result also demonstrates a countervailing effect of anger in strategic interactions. When a group is known to be more sensitive to anger, their aggressive actions are more likely to be forgiven rather than retaliated against. One way to think about the direct and indirect effects of anger in our model is that the direct effects are the effects of anger and the indirect effects are the effects of others’ beliefs about anger. In this sense, it would be unambiguously good for group j if others believed that they are more sensitive to anger. This fits observed deliberate attempts by states to construct certain issues as anger-inducing to preempt confrontation (Hall Reference Hall2011). Since the results demonstrate that an agent has some benefit from being believed by other players to be susceptible to anger, one may ask whether a Friendly group has an ex ante incentive to increase its susceptibility to anger. The answer is no, as Corollary 1 shows.

Corollary 1 follows from the proof of Proposition 3 but we provide a separate proof in the appendix for completeness. The reasoning for why increased susceptibility to anger increases the total probability of conflict despite decreasing the likelihood of retaliation by the other group lies in the relationship between the probability of conflict and the likelihood of retaliation. If a player from group A becomes less likely to observe an aggressive outcome then she becomes more likely to retaliate, both because the aggressive outcome is more informative that the other group may be Hostile and because the aggressive outcome makes her more angry. The same is true for players from group B. What happens then if the value of $ {\alpha}_A $ is increased? Since group A members are more angry, they are now indifferent between conciliatory and aggressive actions at higher values of this probability, which means increasing $ {\alpha}_A $ must increase the likelihood that group A members observe an aggressive outcome. This finding also complements extant empirical research on the consequences of either being too quick to anger (e.g., angry more often) or faking anger in negotiations, which can increase adversary intransigence (Côté, Hideg, and Van Kleef Reference Côté, Hideg and Van Kleef2013; Wong Reference Wong2019), though through different mechanisms than we show here.

The reason for the countervailing effects of anger in our model are also distinct from the existing deterrence logic of anger (Sell, Tooby, and Cosmides Reference Sell, Tooby and Cosmides2009). In our model, it is not just that being susceptible to anger makes one appear “tougher” (Sinaceur and Tiedens Reference Sinaceur and Tiedens2006), unpredictable (McManus Reference McManus2019), less open to compromise (Van Kleef, De Dreu, and Manstead Reference Van Kleef, De Dreu and Manstead2004), or prone to aggression (Fessler Reference Fessler, Potegal, Stemmler and Spielberger2010), but rather, that susceptibility to anger makes it difficult to interpret aggressive actions as indicative of one’s type. If a player is certain that another’s actions are evidence that they are inherently Hostile, they should expect future aggression regardless of their own actions, and so would likely choose aggression. If, however, the player knows that the other is susceptible to anger, choosing a conciliatory response may induce their opponent’s future cooperation.

Another effect of anger in our model is that choices may depend on factors that are irrelevant to material payoffs. For example, Friendly types are more aggressive when $ r(H) $ is larger because increasing $ r(H) $ increases the benefit that a Hostile player receives from a conciliatory action, which reduces the psychological component of that agent’s utility: they do not like to see those who acted aggressively receive a benefit. This cannot happen with no sensitivity to anger: once an agent has the right prediction about how Hostile types will behave, the value of $ r(H) $ is irrelevant to the decision.

Comparison to Naive Anger Preferences

A feature of anger in our model is that cognitive appraisals are anchored to expectations. This assumption is rooted in behavioral research findings but also complicates the analysis. In order to show why modeling anger using psychological game theory produces insights that differ from what could be gained by a simpler approach in which preferences are fully exogenous as in a standard model, we compare our model to one that is identical except for this feature.

Thus, consider the following modification to our basic model. The decision utility for player t is

(3) $$ \begin{array}{rl}{u}_t({x}_t,{\tilde{y}}_{t-1})=\unicode{x1D53C}[{v}_t|{x}_t,{y}_t,{\tilde{y}}_{t-1}]-{\alpha}_j\widehat{\beta}({\tilde{y}}_{t-1})\unicode{x1D53C}[{v}_{t-1}+{v}_{t+1}]& \end{array} $$

where $ \widehat{\beta}(a)=1 $ and $ \widehat{\beta}(c)=0. $ This model, which we refer to as naive anger, requires no deviation from standard game theoretic tools since preferences over outcomes do not depend on conjectures. Rather, players simply get angry whenever they see aggression. However, we highlight one important way in which naive anger cannot capture the same behavior as the more empirically grounded conceptualization of context-dependent anger in our model.

Proposition 4 shows that certain conflict can occur with naive anger for high enough values of $ {\alpha}_A $ and $ {\alpha}_B. $ As Proposition 2 showed, this is not possible in our main model. Thus, the model of context-dependent anger reaches a different conclusion than the naive anger model on the question of whether anger can cause permanent intractable conflict. Our model is thus more realistic on this point: as Acemoglu and Wolitzky (Reference Acemoglu and Wolitzky2014) discuss, even most long and ongoing conflicts are better described by periods of aggression and non-aggression than by constant aggression. This is supported by empirical work on long term rivalries and civil wars. For instance, Goertz, Jones, and Diehl (Reference Goertz, Jones and Diehl2005) emphasize a “punctuated equilibrium” pattern to conflicts with significant wait times in between conflicts which depend on the history of the relationship. It is also common to use hazard models to predict the duration of peaceful spells between episodes of conflict recurrence in civil war settings (Mason et al. Reference Mason, Gurses, Brandt and Quinn2011).

More Information about History

In our main model, we assumed agents have relatively poor information: agents receive a signal of the action taken at time $ t-1 $ and nothing else. This parallels much of the uncertainty that occurs in real-world conflict. However, in some conflicts—such as in longstanding rivalries or recurrent bouts of ethnic conflict—groups may have more information about the past behavior of their adversaries which could shape behavior. For example, research on collective memories in conflict has showcased the importance of long-distant past actions on present conflict attitudes and behavior (Rozenas and Zhukov Reference Rozenas and Zhukov2019; Wayne, Damann, and Fachter Reference Wayne, Damann and Fachter2023).

In Appendix G of the Supplementary Material, we pursue an extension of the model in which agents have more information about the history of play. A key result is that agents’ propensity toward aggression may depend on the history of the game in ways that are not rationalized purely by informational effects. For instance, agents who observe no aggressive actions in the past raise their expectations, which strengthens their anger response when a bad outcome is observed. This may happen even if they are fully convinced that the other group is Friendly. For example, after the Wikileaks scandal that publicized U.S. efforts to spy on its allies, German officials reacted with outrage, asserting “There is a palpable sense of betrayal in Germany over this, across the political spectrum, with calls for retaliatory action […] To many, it feels as if post-war German democracy’s nurturing elder brother, the United States, turned out to be Big Brother” (Busch Reference Busch2014). One interpretation of these events is that knowledge that a group is Friendly (or an ally) does not preclude actors from pursuing an aggressive response out of anger, and may in fact make this action more likely.

Concern for Others’ Beliefs

In our model, anger is based on beliefs about actions: agents are angry when outcomes are worse than expected for reasons attributed to the actions of other players. Another notion of anger may also have agents consider the intentions of other players as part of their cognitive appraisals (Petersen Reference Petersen2010). This means individuals should be particularly angry when they assess the negative actions against them to be unjust or undeserved (Pillutla and Keith Murnighan Reference Pillutla and Keith Murnighan1996) or as violating key values (Hall Reference Hall2017). For instance, in a security dilemma context, leaders may forgive Friendly types’ aggression when they believe it to be defensively motivated due to beliefs the adversary holds about her own actions or preferences, but not if aggression is seen as evidence of an adversary’s inherently Hostile, territorial-aggrandizing preferences (Jervis Reference Jervis1978).

Along these lines, Battigalli, Dufwenberg, and Smith (Reference Battigalli, Dufwenberg and Smith2019) provide a model of “anger blaming intentions” in which agents rely on second-order beliefs. In the anger blaming intentions model, an agent’s anger depends on how much she expects that other players believed that they were reducing her payoff when choosing their action. In our model, we can think about the difference as follows: if I perceive the aggressive actions of the other player to be due to their poor character (e.g., they are a Hostile type), this should make agents angrier than if those same actions are attributed to Friendly types who are simply responding to past aggression. Aggressive action by the other side may be seen as more morally justifiable and therefore less anger-inducing when it is attributed to the other side’s beliefs. Formally, agent t will be less angry when they believe that player $ t-1 $ took an aggressive action believing her to be a Hostile type than if player $ t-1 $ took the same action believing her to be a Friendly type. In Appendix H of the Supplementary Material, we consider this model allowing players’ anger to be reduced to the extent that the action could have been justified by the belief that player t was a Hostile type. The results establish that the main insights from the baseline model also hold for a model in which agents have concern for others’ beliefs, but that the effect of anger are lessened, since players will be potentially more forgiving of aggression.

Informational Effects of Anger

Our models considered the effects of anger on preferences, but the behavioral literature has demonstrated that anger also affects information processing. Angry individuals may become less receptive to new information (Valentino et al. Reference Valentino, Hutchings, Banks and Davis2008), more likely to rely on superficial cues (Bodenhausen, Sheppard, and Kramer Reference Bodenhausen, Sheppard and Kramer1994; Tiedens and Linton Reference Tiedens and Linton2001), and more susceptible to motivated reasoning (Weeks Reference Weeks2015).Footnote ¹⁵ Hence, beliefs are tied to emotions, “where emotion constitutes and strengthens a belief and makes possible a generalization about an actor that involves certainty beyond evidence” (Mercer Reference Mercer2010, 2). Angry individuals are more likely to attribute harmful intent to the actions of others and therefore recommend harsher punishment (Goldberg, Lerner, and Tetlock Reference Goldberg, Lerner and Tetlock1999). Attributing negative outcomes to the actions of others can then induce more anger (Allred Reference Allred, Deutsch and Peter2000). Therefore, anger can become self-reinforcing as angry individuals seek out or process more anger-inducing content (Huddy et al. Reference Huddy, Smirnov, Snider and Perliger2021).

In Appendix I of the Supplementary Material, we incorporate the effects of anger on information processing using the modeling framework of motivated reasoning (Little Reference Little2019; Little, Schnakenberg and Turner, Reference Little, Schnakenberg and Turner2022; Little Reference Little2021), but incorporating a psychological dynamic in which anger-based motivated reasoning is only triggered by a deviation from expected payoffs. We show that the main results apply to this case, indicating that context-dependence and increased aggression are the main features that drive the results, rather than whether anger is more likely to affect preferences or information search.