I Introduction
In resource allocation problems, system planners must make investment decisions to mitigate the risks posed by disturbances or strategic interference. In many practical settings, these investments are made over several stages leading up to the actual time of allocation. For example, security measures in cyberphysical systems are deployed over long periods of time. As such, attackers can use knowledge of predeployed elements to identify vulnerabilities and exploits in the defender’s strategy [3, 23]. As another example, power grid operators must bid on forwardcapacity (i.e., dayahead, hourahead and realtime) markets to fulfill future demand. Although grid operators can significantly reduce energy prices and carbon emissions by procuring capacity in day and hourahead markets, they still rely on realtime markets to account for uncertainty in the demand signal [2, 14]. Further examples include R&D contests, team management in competitive sports, and political lobbying [22].
Indeed, there are numerous realworld examples of systems in which both early and late investments contribute to the system performance. Notably, many of these scenarios consist of strategic interactions between competitors, and exhibit tradeoffs when investing in preallocated and realtime resources (e.g., resource costs vs. flexibility in deployment, longterm vs. shortterm gains). In such scenarios, system planners must choose their dynamic investments while accounting for their competitors’ decision making, and balancing the tradeoffs in early and late investment.
In this manuscript, we seek to characterize the interplay between early and late investment in competitive resource allocation settings. We pursue this research agenda in the context of General Lotto games, a gametheoretic framework that explicitly describes the competitive allocation of resources between opponents. The General Lotto game is a popular variant of the classic Colonel Blotto game, wherein two budgetconstrained players, and , compete over a set of valuable battlefields. The player that deploys more resources to a battlefield wins its associated value, and the objective for each player is to win as much value as possible. Outcomes in the standard formulations are determined by a single, simultaneous allocation of resources. In the novel formulation introduced in this paper, one of the players can strategically decide how to deploy resources before the actual engagement takes place. The placement of the preallocated resources thus has an effect on how the allocation decisions are made at the time of competition.
Specifically, we consider the following twostage scenario. Player is endowed with resources to be preallocated, and both players possess realtime resources to be allocated at the time of competition. In the first stage, player decides how to deterministically deploy the preallocated resources over the battlefields. Player ’s endowments and preallocation decision then become known to player . In the second stage, both players engage in a General Lotto game where they simultaneously decide how to deploy their realtime resources, and payoffs are subsequently derived. We assume player does not have any preallocated resources at its disposal, and only has realtime resources to compete with. Each player can randomize the deployment of her realtime resources. Here, player must overcome both the preallocated and realtime resources deployed by player to secure a battlefield. A full summary of our contributions is provided below:
Our Contributions: Our main contribution in this paper is a full characterization of equilibrium payoffs to both players in our twostage General Lotto game, given player has preallocated resources, realtime resources, and player has realtime resources (Theorem III.1). This result also specifies how player should optimally deploy its preallocated resources to the battlefields, each of which has an arbitrary associated value. Our characterization explicitly reveals the relative effectiveness of preallocated and realtime resources – for any desired performance level against , we provide the set of all pairs that achieve the payoff for player (Theorem III.2). As a consequence, we show that, to achieve the same performance using only one type of resource, player needs at least double the amount of preallocated resources than the amount of realtime resources (Corollary 3.1).
Leveraging the main results, we then derive the optimal investment pair for player when there are linear perunit costs to invest in both types of resources and a limited monetary budget is available. We note that it is optimal to invest in both resources only if the perunit cost of preallocated resources is lower than realtime resources. Indeed, preallocated resources are less effective than realtime resources, since their deployment is not randomized and player has knowledge of their placement.
Related works: This manuscript takes preliminary steps towards developing analytical insights about competitive resource allocation in multistage scenarios. There is widespread interest in this research objective, where the focus ranges from zerosum games [15, 9, 13], and dynamic games [8, 19], to Colonel Blotto games [20, 1, 18, 12]. The goal of many of these works is to develop computational tools to compute decisionmaking policies for agents in adversarial and uncertain environments. In comparison, our work provides explicit, analytical characterizations of equilibrium strategies, allowing for insights relating the players’ performance with various elements of adversarial interaction to be drawn. As such, our work is related to a recent research thread studying Colonel Blotto games in which allocation decisions are made over multiple stages [10, 7, 6, 16, 4].
Our work also draws significantly from the primary literature on Colonel Blotto and General Lotto games [5, 17, 11, 21]. In particular, the simultaneousmove subgame played in the second stage of our model was first proposed by Vu and Loiseau [21], and is known as the General Lotto game with favoritism (GLF). Favoritism refers to the fact that preallocated resources provide an inherent advantage to one player’s competitive chances. Their work establishes existence of equilibria and develops computational methods to calculate them to arbitrary precision. However, this prior work considers preallocated resources as exogenous parameters of the game. In contrast, we model the deployment of preallocated resources as a strategic element of the competitive interaction. Furthermore, we provide the first analytical characterization of equilibria and the corresponding payoffs in GLF games.
Ii Problem formulation
The General Lotto game with preallocations (GLP) is a twostage game with players and , who compete over a set of battlefields, denoted as . Each battlefield is associated with a known valuation , which is common to both players. Player is endowed with a preallocated resource budget and a realtime resource budget . Player is endowed with a realtime resource budget , but no preallocated resources.^{1}^{1}1Recent computational advances (see, e.g., [21]) permit the study of the scenario where both players are endowed with preallocated resources. In this work, we seek to provide analytical characterizations of equilibrium payoffs, and, thus, consider the simpler, unilateral preallocation setting. The two stages are played as follows:
– Stage 1: Player decides how to allocate her
preallocated resources to the battlefields, i.e., it selects a vector
. We term the vector as player ’s preallocation profile. No payoffs are derived in Stage 1, and ’s choice becomes binding and common knowledge.– Stage 2: Players and compete in a simultaneousmove subgame over with their realtime resource budgets , . Here, both players can randomly allocate these resources as long as their expenditure does not exceed their budgets in expectation. Specifically, a strategy for player is an variate (cumulative) distribution over allocations that satisfies
(1) 
We use to denote the set of all strategies that satisfy (1). Given that player chose in Stage 1, the expected payoff to player is given by
(2) 
where if , and otherwise for any two numbers .^{2}^{2}2The tiebreaking rule (i.e., deciding who wins if ) can be assumed to be arbitrary, without affecting any of our results. This property is common in the General Lotto literature, see, e.g., [11, 21]. In words, player must overcome player ’s preallocated resources as well as player ’s allocation of realtime resources in order to win battlefield . The parameter is the relative quality of player ’s realtime resources against player ’s resources. When (resp. ), they are less (resp. more) effective than player ’s resources. The payoff to player is , where we denote .
Stages 1 and 2 of GLP are illustrated in Figure 1a. We denote an instance of GLP as , and note that the Stage 2 subgame (i.e., the game with fixed preallocation profile) is an instance of the General Lotto game with favoritism [21]. For a given GLP instance , we define an equilibrium as any joint strategy profile that satisfies
(3)  
for any , and . Notably, player ’s strategy consists of her deterministic preallocation profile in Stage 1, as well as her randomized allocation of realtime resources in Stage 2. It follows from the results in [21] that an equilibrium exists in any GLP instance , and that the equilibrium payoffs , , are unique. For ease of notation, we will use , , to denote players’ equilibrium payoffs in when the dependence on the vector is clear.
Iii Main results
In this section, we present our main result: the characterization of players’ equilibrium payoffs in the GLP game. We then use this result to derive an expression for the level sets of the function in , and to compare the relative effectiveness of preallocated and realtime resources.
The result below provides an explicit characterization of player ’s equilibrium payoff . Note that player ’s equilibrium payoff is simply .
Theorem Iii.1.
Consider a GLP game instance with , and . The following conditions characterize player ’s equilibrium payoff :

If , or and , then is
(4) 
Otherwise, is
(5)
The derivation of the above result is challenging because explicit expressions for the players’ payoffs in the Stage 2 subgame are generally not attainable for arbitrary . Moreover, these payoffs are not generally concave. Our approach is to show that for any , the payoff is nondecreasing in the direction pointing to . The full proof is given in Appendix B, and relies on methods developed in [21]. These details are given in Appendix A.
As a consequence of our main result, we are able to characterize expressions for the level curves of the function . That is, for a desired performance level and fixed , we provide the set of all pairs such that .
Theorem Iii.2.
Given any and , fix a desired performance level . The set of all pairs that satisfy is given by the following conditions:
If , then
(6) 
If , then
(7) 
If , then for any .
We plot the surface for as well as the level curves corresponding to in Figure 1b. Notably, for any , the level curve is strictly decreasing and convex in , where we use to explicitly note the dependence on . Hence, the function is quasiconcave in .
We can use the result in Theorem III.2 to obtain an expression for the relative effectiveness of preallocated and realtime resources when these are deployed in isolation. In the following corollary, we provide this expression, and observe that realtime resources are at least twice as valuable as preallocated resources, and can be arbitrarily more valuable in specific settings:
Corollary 3.1.
For given , the unique value such that is characterized by the following expression:
(8) 
Notably, the ratio is lowerbounded by , and as .
Iv Optimal investment decisions
In this section, we consider a scenario where player has an opportunity to make an investment decision regarding its resource endowments. That is, the pair is a strategic choice made by before the game is played. Given a monetary budget for player , any pair must belong to the following set of feasible investments:
(9) 
where is the perunit cost for purchasing preallocated resources, and we assume the perunit cost for purchasing realtime resources is 1 without loss of generality. We are interested in characterizing player ’s optimal investment subject to the above cost constraint, and given player ’s resource endowment . This is formulated as the following optimization problem:
(10) 
In the result below, we derive the complete solution to the optimal investment problem (10).
Corollary 4.1.
Fix a monetary budget , relative perunit cost , and realtime resources for player . Then, player ’s optimal investment in preallocated resources for the optimization problem in (10) under the linear cost constraint in (9) is
(11) 
where . The optimal investment in realtime resources is . The resulting payoff to player is given by
(12) 
The above solution is obtained by leveraging the level set characterization from Theorem III.2, and the fact that the level sets are strictly decreasing and convex for . We omit details of the proof for space considerations. A visual illustration of how the optimal investments are determined is shown in Figure 2. The budget constraint is a line segment in , and we thus seek the level curve that lies tangent to the segment. In cases where the cost is sufficiently high, no level curve lies tangent to , and, thus, player invests all of her budget in realtime resources.
V Conclusion
In this manuscript, we studied the strategic role of preallocations in competitive interactions under a twostage General Lotto game model. We identified an explicit expression for the set of preallocated and realtime budget pairs that correspond with a given desired performance. We then used this explicit expression to derive the optimal dynamic investment strategy under a given linear cost constraint, and to compare the relative effectiveness of preallocated and realtime resources when deployed in isolation. Exciting directions for future work include studying the strategic outcomes (i.e., equilibria) when both players can make preallocations, and introducing heterogeneities in players’ battlefield valuations and resource effectiveness to the model.
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