1  In line with orthodox economic terminology, type refers to payoffrelevant parameters. 
2  In a cursed equilibrium, players draw partial inference about the correlation between other players’ actions and their types. Informational naivety is the fully cursed equilibrium, whereby no inference is drawn. Informational naivety is also a special case of an analogybased expectation equilibria, where players’ analogy partitions coincide with their own information partitions (see Eyster and Rabin, 2010, p. 1634 [ 14]; Jehiel and Koessler, 2007, p. 539 [ 15]; and Ettinger and Jehiel, 2010 [ 16], footnote 7). Finally, informational naivety is a special case of selfconfirming equilibria, where the cursed players observe only the aggregate play of the opponents and neither the state nor the opponent’s type (Fudenberg, 2006) [ 17]. The three concepts differ in how they convexify informational naivety and informational sophistication. Since we focus only on these two extremes, we do not need to take a stand among the three concepts. 
3  In the general model of Section 3, we will only adopt the weaker assumption, $\left(MonSpoils\right)$, that the spoils of war increase in the rival’s resources. 
4  For instance, Jackson and Morelli (2007) [ 18] assume that the probability of winning the war satisfies $\left(MonProb\right)$ without explicitly modeling the arming phase. Bueno de Mesquita (1981, p. 102) [ 19] also assumes that wealth translates into military capability. Hörner et al.’s (2015) [ 20] workhorse model has two types, h and l, and probabilities satisfy ${p}_{h,l}>1/2={p}_{h,h}={p}_{l,l}>{p}_{l,h}$, consistent with $\left(MonProb\right)$. Taking types as sunk military investments, Meirowitz and Sartori (2008) [ 21] assume $\left(MonProb\right)$. Furthermore, $\left(MonProb\right)$ typically arises in conflict models with resource constraints, such as Tullock contests and Colonel Blotto games. For empirical evidence supporting $\left(MonProb\right)$, see, for instance, footnote 15 in Jackson and Morelli (2007) [ 18] and the references therein. 
5  The normalization of low resources to 0 is qualitatively innocuous. 
6  The result would carry over if the costs of war $c\ge 0$ depend on players’ resources or are paid by both players rather than only by the winner. Similarly, the normalization of the loser’s payoff to 0 is without loss of generality, and for simplicity, the normalization is maintained throughout the paper. 
7  Peace payoffs depend on resources but not on probabilities of victory and are thus unaffected by informational naivety. 
8  The conditions corresponding to Equations ( 1) and ( 2) for a player with 0 resources read as follows:

9  Notice that, while for 0resource players, $0\le {c}_{0}^{IS}<{c}_{0}^{IN}$, for Rresource players, ${c}_{R}^{IS}\le 0$ and ${c}_{R}^{IN}\ge 0\iff p\ge 5/6$. Hence, while the intermediate range of costs always exists for 0resource players, it may not exist for Rresource players. 
10  An interpretation is that, if a player wins, they gain only a fixed fraction of the rival’s resources, as assumed, for instance, by Jackson and Morelli (2007) [ 18]. 
11  Throughout the paper, we omit, for the sake of space, the dependence of $f\left({\theta}_{i},{\theta}_{j}\right)$ on any variable different than types. Recall, however, that $f\left({\theta}_{i},{\theta}_{j}\right)$ may depend on any number of other parameters, such as, in the examples spelled out so far, the destructiveness parameters ${d}_{1},{d}_{2}$, and the cost of war c. 
12  Note that we allow for $p\left({\theta}_{i},{\theta}_{i}\right)\ne 1/2$. As pointed out by Jackson and Morelli (2007) [ 18], “[T]his allows i, for instance, to have some geographic, population, or technological advantage or disadvantage.” 
13  Notice that, when $i=n$ ($i=1$), the first (second) summation within the square bracket is null. 
14  The allpay auction technology, where ${p}_{i}\left({\theta}_{i},{\theta}_{j}\right)=1$ when ${\theta}_{i}>{\theta}_{j}$, ${p}_{i}\left({\theta}_{i},{\theta}_{j}\right)=0$ when ${\theta}_{i}<{\theta}_{j}$, and ${p}_{i}\left({\theta}_{i},{\theta}_{j}\right)=1/2$ when ${\theta}_{i}={\theta}_{j}$, satisfies the weakening of $\left(MonProb\right)$, but for some pairs of types, it may not satisfy the original $\left(MonProb\right)$. 
15  The function $f:{\Theta}^{2}\to \Re $ satisfies strict supermodularity in $\left({\theta}_{i},{\theta}_{j}\right)$ if $f\left({\theta}_{i}^{\prime},{\theta}_{j}^{\prime}\right)f\left({\theta}_{i}^{\prime},{\theta}_{j}\right)>f\left({\theta}_{i},{\theta}_{j}^{\prime}\right)f\left({\theta}_{i},{\theta}_{j}\right)$ for any ${\theta}_{i}^{\prime}>{\theta}_{i}$ and ${\theta}_{j}^{\prime}>{\theta}_{j}$. Strict submodularity is similarly defined. 
16  Knifeedge as $f\left({\theta}_{i}^{\prime},{\theta}_{j}^{\prime}\right)f\left({\theta}_{i}^{\prime},{\theta}_{j}\right)=f\left({\theta}_{i},{\theta}_{j}^{\prime}\right)f\left({\theta}_{i},{\theta}_{j}\right)$. 
17  If both players exert 0 effort, each player has $1/2$ probability of victory. 
18  Throughout this subsection and the next one, we spell out the maximization problem of player 1. That of player 2 is symmetric. 
19  Throughout this subsection and the next one, the secondorder conditions (SOCs) hold. 
20  Steps identical to the proof of Lemma 1 would also show that ${e}^{R}>{e}^{0}$ in any interior typesymmetric equilibrium. 
21  For pioneering works, see, for instance, Bueno de Mesquita and Lalman (1986) [ 26], Morrow (1989) [ 4], and Banks (1990) [ 27]. 
22  A similar argument could be made for a systematic overestimation of the spoils of war: subjective estimations of the spoils of war sum to more than the objective stakes. 
23  When bidders share a common but unknown value for the object at auction and receive private signals about such value, they tend to bid more than equilibrium theory predicts; this phenomenon is known as the winner’s curse. A bidder wins if others bid sufficiently lower than them, which happens if others received private signals that are more negative than their own. A bidder who fails to draw such an inference between others’ bids (actions) and signals (types) overestimates the value of the object and, hence, overbids (e.g., Kagel and Levin, 1986 [ 3]; Eyster and Rabin, 2005 [ 11]; and the references therein). 
24  Subjects disproportionately enter competitions on easy tasks (Moore and Cain, 2007) [ 32]. eBay sellers disproportionately choose to end their auctions during the times of day when more bidders are online (Simonsohn, 2010) [ 33]. Informationally naïve people herd with positive probability on incorrect actions (Eyster and Rabin, 2010) [ 14]. 
25  If a prospective buyer fails to realize that a bid will only be accepted if the seller’s valuation is less than the bid, then the buyer might incur a loss (e.g., Holt and Sherman, 1994) [ 10]. 
26  See Lindsey (2018) [ 34] for a recent contribution overcoming the issue of the difficulty of observing country leaders’ beliefs. See also Lai (2004) [ 35] and Bas and Schub (2016) [ 36] for similar empirical evidence. For laboratory experiments directly manipulating information, see Tingley and Wang (2010) [ 37] and Quek (2015) [ 38]. 