Approved for public release; distribution in unlimited. / In this thesis, a new fictitious play (FP) procedure is presented to solve two-person zero-sum (TPZS) Blotto games. The FP solution procedure solves TPZS games by assuming that the two players take turns selecting optimal responses to the opponent's strategy observed so far. It is known that FP converges to an optimal solution, and it may be the only realistic approach to solve large games. The algorithm uses dynamic programming (DP) to solve FP subproblems. Efficiency is obtained by limiting the growth of the DP state space. Blotto games are frequently used to solve simple missile defense problems. While it may be unlikely that the models presented in this paper can be used directly to solve realistic offense and defense problems, it is hoped that they will provide insight into the basic structure of optimal and near-optimal solutions to these important, large games, and provide a foundation for solution of more realistic, and more complex, problems. / Captain, Republic of Korea Air Force

<p>Contest outcomes often involve some mix of skill and chance.  In three essays, I vary the sources of noise and show how player actions either influence, or are influenced by, noise.  I begin with a classic multi-battle contest, the Colonel Blotto game.  Due to his disadvantage in resources, the weak player in this contest stochastically distributes resources to a subset of battlefields while neglecting all others in an attempt to achieve a positive payoff.  In contrast, the strong player evenly distributes his resources in order to defend all battlefields, while randomly assigning extra resources to some.  Because the weak player benefits from randomizing over larger numbers of battlefields, a strong player has incentive to decrease the range over which the weak player can randomize.  When battlefields are exogenously partitioned into subsets, or \textit{fronts}, he is able to do this by decentralizing his forces to each front in a stage prior to the distribution of forces to battlefields and actual conflict.  These allocations are permanent, and each subset of battlefields effectively becomes its own, independent Blotto subgame.  I show that there exist parameter regions in which the strong player's unique equilibrium payoffs with decentralization are strictly higher than the unique equilibrium payoffs without decentralization.</p> <p><br></p> <p>In my second paper, I show how sources of exogenous noise, what Clausewitz referred to as the ``fog of war," obscure developments on the battlefield from the view of a military leader, while individual inexperience and lack of expertise in a particular situation influence his decisionmaking.  I model both forms of uncertainty using the decentralized Colonel Blotto game from the first chapter.  To do so, I first test the robustness of allocation-stage subgame perfect equilibria by changing the contest success function to a lottery, then I find the players' quantal response equilibria (QRE) to show how individual decision-making is impacted by bounded rationality and noisy best responses, represented by a range of psi values in the logit QRE.  I find that player actions rely significantly less on decentralization strategies under the lottery CSF compared to the case of the all-pay auction, owing mainly to the increased exogenous noise.  Moreover, agent QRE and heterogeneous QRE approximate subgame perfect equilibria for high values of psi in the case of an all-pay auction, but under the lottery CSF, QRE is largely unresponsive to changes in psi due to the increase in exogenous noise.</p> <p><br></p> <p>Finally, I examine a potential method for introducing noise into the all-pay auction (APA) contest success function (CSF) utilized in the Colonel Blotto games of the first two chapters.  Many contests are fundamentally structured as APA, yet there is a tendency in the empirical literature to utilize a lottery CSF when stochastic outcomes are possible or the tractability of pure strategy equilibria is desired.  However, previous literature has shown that using a lottery CSF sacrifices multiple distinguishing characteristics of the APA, such as the mixed strategy equilibria described by Baye, Kovenock, and de Vries (1996), the exclusion principle of Baye, Kovenock, and de Vries (1993), and the caps on lobbying principle of Che and Gale (1998).  I overcome this by formulating an APA that incorporates noise and retains the defining characteristics of an auction by forming a convex combination of the APA and fair lottery with the risk parameter lambda.  I prove that equilibria hold by following the proofs of Baye et al. (1996, 1993) and Che and Gale (1998), and I show the new CSF satisfies the axioms of Skaperdas (1996).  While player and auctioneer actions, payments, and revenues in the noisy APA adhere closely to the those of the APA for low levels of noise, the effect of discounted expected payoffs results in lower aggregate payments and payoffs when noise is high.  Finally, I show the noisy APA is only noise equivalent to the lottery CSF when lambda = 0, i.e., the fair lottery.</p>