Strategic betting for competitive agents

In many multiagent settings, each agent's goal is to come out ahead of the other agents on some metric, such as the currency obtained by the agent. In such settings, it is not appropriate for an agent to try to maximize its expected score on the metric; rather, the agent should maximize its expected probability of winning. In principle, given this objective, the game can be solved using game-theoretic techniques. However, most games of interest are far too large and complex to solve exactly. To get some intuition as to what an optimal strategy in such games should look like, we introduce a simplified game that captures some of their key aspects, and solve it (and several variants) exactly. Specifically, the basic game that we study is the following: each agent i chooses a lottery over nonnegative numbers whose expectation is equal to its budget bi. The agent with the highest realized outcome wins (and agents only care about winning). We show that there is a unique symmetric equilibrium when budgets are equal. We proceed to study and solve extensions, including settings where agents must obtain a minimum outcome to win; where agents choose their budgets (at a cost); and where budgets are private information.