Strategically revealing intentions in General Lotto games

TL;DR

The paper analyzes whether publicly revealing strategic intentions via pre-commitments can improve outcomes in General Lotto games. By modeling a sequenced, three-stage interaction and separating symmetric and asymmetric battlefield valuations, it derives precise budget- and value-based conditions under which pre-commitments are beneficial, including results that a single high-value battlefield pre-commitment often dominates multi-battlefield pre-commitments. It shows that in symmetric valuations, a weaker player has no incentive to pre-commit, while a stronger player can gain under certain thresholds; in asymmetric valuations, even weaker players can benefit by pre-committing, particularly to induce withdrawals and beat the second-best nominal equilibria. Collectively, the findings reveal that strategic signaling can improve payoffs in adversarial resource allocation, with implications for defense planning and mechanism design in security-sensitive domains.

Abstract

Strategic decision-making in uncertain and adversarial environments is crucial for the security of modern systems and infrastructures. A salient feature of many optimal decision-making policies is a level of unpredictability, or randomness, which helps to keep an adversary uncertain about the system's behavior. This paper seeks to explore decision-making policies on the other end of the spectrum -- namely, whether there are benefits in revealing one's strategic intentions to an opponent before engaging in competition. We study these scenarios in a well-studied model of competitive resource allocation problem known as General Lotto games. In the classic formulation, two competing players simultaneously allocate their assets to a set of battlefields, and the resulting payoffs are derived in a zero-sum fashion. Here, we consider a multi-step extension where one of the players has the option to publicly pre-commit assets in a binding fashion to battlefields before play begins. In response, the opponent decides which of these battlefields to secure (or abandon) by matching the pre-commitment with its own assets. They then engage in a General Lotto game over the remaining set of battlefields. Interestingly, this paper highlights many scenarios where strategically revealing intentions can actually significantly improve one's payoff. This runs contrary to the conventional wisdom that randomness should be a central component of decision-making in adversarial environments.

Figures (6)

• Figure 1: Model of public pre-commitments. (a) For simplicity, we depict General Lotto games where battlefield valuations are symmetric across players in these diagrams, i.e. $\text{GL}(X_A,X_B,\boldsymbol{v})$. This is the nominal interaction if no player has the option to pre-commit resources, or the pre-committing player chooses not to pre-commit any resources. (b) Diagrams showing the sequence of events when player $B$ has the option to pre-commit. In step 1, $B$ pre-commits $p_1$ and $p_2$ resources to battlefields 1 and 2, respectively. In step 2, $A$ publicly responds to the pre-commitments by either matching or withdrawing. It secures the battlefields it matches, and loses the ones it withdraws from. In step 3, the players engage in a GL game on the remaining set of battlefields with their remaining resources.
• Figure 2: Scenario 1: Symmetric valuations. (a) Player $B$ pre-commits to a single battlefield of its choice. Our results establish that a pre-commitment to a single battlefield is preferable over pre-commitments to multiple battlefields of the same total value (Lemma \ref{['lem:single_t_optimal']}). We thus seek necessary and sufficient conditions for which pre-committing to a single battlefield is beneficial for some set of valuations $\boldsymbol{v}$. (b) Parameter regimes where a player has an incentive to pre-commit (blue regions), i.e. to outperform its equilibrium payoff in the nominal GL game (Result 1). In this example, the value of the battlefield that $B$ can pre-commit to is restricted to not exceed 0.55. Here, we consider any set of valuations $\boldsymbol{v}$ whose total value is fixed to one. A full characterization of regions is given in Theorem \ref{['thm:GL_result']}. Incentives only exist when the pre-committing player is stronger than its opponent. (c) The percent improvement in payoff that pre-commitments offer to player $B$ over playing the nominal GL game, as a function of the enemy's budget $X_A$ (tracing out parameters of the dotted vertical line in (b)). Here, the pre-committing player's budget is fixed to $X_B=3$. Pre-committing can never outperform the nominal payoff when the player is weaker ($X_B < X_A$).
• Figure 3: Scenario 2: Asymmetric valuations. (a) The left diagram is the game $\text{GGL}(X_A,X_B,\alpha)$ under consideration. The right diagram shows the scenario where $B$ has the option to pre-commit resources to a battlefield. After the opponent's response, they play a GL game on the remaining battlefield. (b) Parameter regions where a player in the GGL game has an incentive to pre-commit, based on two different criteria. The green region indicates parameters where there is a unique equilibrium payoff of the underlying GGL game, and there are beneficial pre-commitments that exceed this payoff. In the blue region, the GGL game admits multiple equilibria, and pre-commitments can exceed the second-highest equilibrium payoff (Theorem \ref{['thm:second_best']}). Multiple equilibria arise in the red region, but no such pre-commitments exist. Observe that in this setting, incentives exist for a weaker player (blue region), whereas this was never the case for symmetric valuations. (c) The equilibrium payoffs to player $B$ in the nominal (no pre-commitments) GGL game (traces out parameters of the dotted vertical line in (b)). The GGL game admits three equilibria in the blue and red regions, and the payoffs are ranked from best to worst for $B$. The dashed green line indicates the best payoff attainable from a pre-commitment. In this example, we note that it exceeds the best equilibrium payoff by an insignificant margin for $X_A \in [0,2.5]$ (approximately). However, it can improve upon the second-highest equilibrium payoff by up to 20$\%$.
• Figure 4: (a) Full parameter region (in green) where there exists beneficial pre-commitments in $\text{GL}(X_A,X_B,\boldsymbol{v})$ (from Theorem \ref{['thm:GL_result']}). Here, we set $\phi=1$. (b,c) These diagrams depict regions of player A's optimal response against any pre-commitment $p \in [0,X_B]$ on a single battlefield of value $v_b \in [0,\phi]$ (suppose $\bar{v} = \phi$ here), and the total value of battlefields is $\phi$. (b) When $X_A<X_B<2X_A$, there is an incentive to pre-commit to a single battlefield $b$ if and only if $v_b > (1 - \frac{X_A/X_B}{3-X_B/X_A})\phi$ (green region). The set of beneficial pre-commitments are shown as the shaded green region. The exact analytical characterizations of the border lines are given in the proof of Theorem \ref{['thm:GL_result']} (Appendix).
• Figure 5: (a) Diagram of our simulation setup. First, a random set of valuations $\boldsymbol{v} \in \mathcal{V}$ ($n=3$) is drawn from the uniform distribution on $\mathcal{V}$. We calculate the optimal two-battlefield pre-commitment to the first two battlefields. We then calculated the optimal pre-commitment to a single battlefield of value $v_1+v_2$. Here, $X_B = 1.5$ and $X_A = 1$. (b) We performed calculations on 500 independent samples of $\boldsymbol{v}$ (for each $\bar{v}$ ranging from 0:0.05:1), with $\phi = 1$ and $n=3$. This plot shows the average payoff obtained from pre-committing to a single battlefield (blue) vs pre-committing to two battlefields (red). (c) This plot shows the percent of samples for which the optimal pre-commitment provided benefits over the nominal payoff. Note that no benefits are available for any $\bar{v} < 5/9$ (Theorem \ref{['thm:GL_result']}).

Theorems & Definitions (13)

• Definition 1
• Lemma 2.1
• proof
• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2