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Forming Alliances in the Fog of War: A General Lotto Perspective

Edik Hakobyan, Vade Shah, Jason R. Marden

arXiv:2610.00608v1cs.GT

TL;DR

This paper asks whether alliances remain worthwhile when players know relevant budgets only up to a multiplicative factor. It analyzes uncertainty about either the adversary’s budget or the ally’s budget, characterizing robust mutually beneficial transfers and showing that they exist in a nontrivial subset of games at every uncertainty level.

  • Problem

    Whether mutually beneficial budget transfers remain worthwhile when players have uncertain knowledge of the adversary’s or ally’s budget is an open question.

    Source: §I · pp. 1–2

  • Method

    The paper analyzes two coalitional General Lotto models, separately treating uncertainty about the adversary’s budget and about the other player’s budget.

    Source: B. Coalitional General Lotto Game · p. 2, §I · p. 2

  • Results

    For every uncertainty level, each model has a nontrivial subset of games admitting robust mutually beneficial transfers, which the paper characterizes exactly.

    Source: §I · p. 2, A. Adversary Uncertainty · p. 4, B. Ally Uncertainty · p. 5

  • Takeaways & Limitations

    Alliances can remain worthwhile under budget uncertainty in some games, even when players lack precise budget knowledge.

    Source: §IV · p. 6

Abstract

from arXiv · show

Agents competing against a common opponent can often gain an edge by forming alliances, but deciding whether to do so is difficult when they lack precise knowledge about their opponent or ally. We study how uncertainty affects opportunities for alliance formation in the coalitional General Lotto game, in which two players compete aganist a shared adversary by allocating their resource budgets over separate sets of valued contests. When all agents know one another's budgets, it is known that a budget transfer from one player to the other can benefit both players. We ask whether such alliances survive when the players know the relevant budgets only up to a multiplicative factor. Under two models, one in which the players are uncertain about the adversary's budget and one in which they are uncertain about each other's budgets, we characterize exactly which games admit a transfer that benefits both players in every game consistent with what they conjecture to be possible. In both models, we show that such robust transfers exist in a set of games of positive measure for every level of uncertainty. These results suggest that alliances can remain worthwhile even when allies are largely in the dark.

I. INTRODUCTION

Alliances can strengthen agents competing against a shared opponent, but uncertainty about opponents’ or allies’ resources makes their value harder to assess. This paper studies whether transfers remain mutually beneficial under uncertainty about the adversary’s budget or the other player’s budget.

I. INTRODUCTION

  • Agents may form alliances against shared opponents by pooling or transferring resources to address vulnerabilities and build on one another’s strengths.The introduction illustrates this motivation with companies, energy producers, and countries.Source: §I · p. 1
  • In General Lotto, agents allocate limited resources across valued contests, aiming to maximize the value won while meeting their budgets in expectation.The agent committing more resources to a contest wins its value.Source: §I · p. 1
  • In the coalitional game, two players face a shared adversary in separate contests, and a transfer can benefit both by prompting the adversary to reallocate resources.Prior work established such mutually beneficial transfers when budgets are known exactly.Source: §I · p. 1
  • The open question is whether a transfer that helps both players under exact budget knowledge remains worthwhile when a budget is misjudged.The introduction distinguishes this question from prior work on uncertainty in one-versus-one General Lotto games.Source: §I · p. 1
  • For both uncertainty models, robust mutually beneficial transfers exist in a nontrivial subset of games at every uncertainty level, and the paper characterizes when they do.The models concern uncertainty about the adversary’s budget and uncertainty about the ally’s budget.Source: §I · p. 2

II. MODEL

The section introduces the coalitional General Lotto model and first reviews the classical one-player-versus-adversary General Lotto game.

II. MODEL

  • The model section begins with the classical one-versus-one General Lotto game before introducing the coalitional version.The coalitional game has two players independently competing against a common adversary across disjoint contest sets.Source: §II · p. 2

A. General Lotto Game

General Lotto models two agents allocating divisible budgets across valued contests, with the higher allocation winning each contest. Equilibrium payoffs are characterized for this setting.

A. General Lotto Game

  • The Player and Adversary allocate arbitrarily divisible budgets across contests, and the higher allocation wins each contest’s value, with ties awarded to the Player.Budgets and contest valuations are common knowledge, although neither agent knows the opponent’s allocation.Source: A. General Lotto Game · p. 2
  • The Player’s strategy is a distribution over allocations that meets the budget constraint in expectation, and the Adversary’s strategy is defined similarly.The section presents equilibrium payoffs rather than the equilibrium strategies.Source: A. General Lotto Game · p. 2
  • For any General Lotto game with budgets XP and XA and contest valuations v1, ..., vn, a Nash equilibrium exists, and the Player’s and Adversary’s equilibrium payoffs are unique.Source: A. General Lotto Game · p. 2

B. Coalitional General Lotto Game

The coalitional General Lotto game has two players compete against a shared adversary in separate contests, with a unilateral resource transfer forming an alliance. The staged interaction lets the adversary respond to transfers before equilibrium allocations and payoffs are realized.

B. Coalitional General Lotto Game

  • An alliance is formed through a unilateral transfer between players competing against a common adversary in separate General Lotto games.The game is parameterized by the players’ and adversary’s budgets and the cumulative contest valuations in each game.Source: B. Coalitional General Lotto Game · p. 2
  • The game proceeds from initialization to player transfers, then adversary best response, and finally equilibrium resource allocation and payoffs.Figure 2 depicts the transfer before the adversary responds.Source: B. Coalitional General Lotto Game · pp. 2–3, Figure 2 · p. 3
  • A transfer is mutually beneficial when it strictly increases both players’ payoffs relative to no transfer.The solution pair is unique except at one transfer, a measure-zero case where the adversary may be indifferent among allocations yielding different player payoffs.Source: B. Coalitional General Lotto Game · p. 3
  • Prior work found mutually beneficial transfers in a nontrivial subset of games, with any such transfer flowing from Player 2 to Player 1 under the stated budget-to-valuation ordering.The analysis focuses on games where Player 2’s budget-to-valuation ratio is at least Player 1’s; results extend to the opposite ordering by swapping indices.Source: B. Coalitional General Lotto Game · p. 3

III. RESULTS

This section examines whether mutually beneficial budget transfers persist when players are uncertain about game parameters. It considers two uncertainty models and focuses on games where, without uncertainty, beneficial transfers flow from Player 2 to Player 1.

  • The analysis asks whether budget transfers can remain mutually beneficial when Players are uncertain about game parameters.Source: §III · p. 3
  • The section considers two uncertainty models for assessing whether mutually beneficial transfers persist.Source: §III · p. 3
  • Throughout, the analysis assumes games satisfying (11), where any mutually beneficial transfer without uncertainty flows from Player 2 to Player 1.For convenience, G |P 7→P ′ denotes a game identical to G except that P is replaced by P ′.Source: §III · p. 3

A. Adversary Uncertainty

Under uncertainty about the Adversary’s budget, the paper characterizes when mutually beneficial transfers remain robust. For every uncertainty level δ ∈ [0, 1), such transfers exist in a positive-measure set of games.

A. Adversary Uncertainty

  • Players observe an uncertainty set for the Adversary’s budget, and a transfer is robust only if it benefits both Players across the games in that set.Source: A. Adversary Uncertainty · pp. 3–4
  • At δ = 0.4, the illustrated game has no robust transfer under Adversary uncertainty, although the nominal game has a mutually beneficial transfer.The figure also contrasts this with persistence under ally uncertainty.Source: Figure 3 · p. 4, A. Adversary Uncertainty · p. 4
  • Theorem 3 gives a complete characterization of games admitting robust mutually beneficial transfers under Adversary uncertainty.The supplied theorem passages do not include the full conditions.Source: A. Adversary Uncertainty · p. 4
  • For every δ ∈ [0, 1), a positive-measure set of games admits a transfer robust to Adversary uncertainty.The paper presents this as the significance of its characterization.Source: A. Adversary Uncertainty · p. 4
  • The result indicates that opportunities for alliance formation do not disappear entirely at any uncertainty level considered, including when Players have little information about the Adversary’s resources.Source: A. Adversary Uncertainty · p. 4

B. Ally Uncertainty

Under ally uncertainty, the paper characterizes when mutually beneficial transfers remain robust. For every δ ∈ [0, 1), these transfers exist in a positive-measure set of games.

B. Ally Uncertainty

  • Each Player knows the other’s budget only within a multiplicative uncertainty set, and the Players do not know one another’s uncertainty sets.The model assumes a common δ.Source: B. Ally Uncertainty · p. 5
  • A proposed transfer can reveal budget-ratio information: the giving Player must be certain their post-transfer ratio remains higher, and the recipient can infer a ratio ordering.Source: B. Ally Uncertainty · p. 5
  • Theorem 5 characterizes games admitting mutually beneficial transfers robust to ally uncertainty.The supplied theorem passages do not include the complete conditions.Source: B. Ally Uncertainty · p. 5
  • For every δ ∈ [0, 1), a positive-measure set of games admits a transfer robust to ally uncertainty.The paper presents this as the significance of Theorem 5.Source: B. Ally Uncertainty · p. 5
  • The illustrated game admits a transfer robust to ally uncertainty at δ = 0.6.Figure 5 depicts games in the X1–X2 plane for several uncertainty levels.Source: Figure 5 · p. 6
  • The paper concludes that opportunities do not vanish entirely even when Players have almost no information about one another’s resource levels.Source: B. Ally Uncertainty · pp. 5–6

IV. CONCLUSION

Robust mutually beneficial transfers exist in a nontrivial subset of games at every uncertainty level in both uncertainty models. The adversary’s best response is described through cases that determine how its budget is allocated across the two games.

IV. CONCLUSION

  • For any uncertainty level, both adversary-budget and ally-budget uncertainty admit robust mutually beneficial transfers in a nontrivial subset of games.The paper characterizes exactly when these alliances remain mutually beneficial in each model.Source: §IV · p. 6
  • The conclusion calls for extending the analysis to more general competitive networks, including many allies or multiple adversaries.Source: §IV · p. 6

A. Adversary’s Best Response

B. Results for the Nominal Game

The nominal-game results relate the transfer conditions across adversary-response cases and establish when a transfer benefits both players. In particular, a giver’s gain implies a receiver’s gain, and the relevant condition remains equivalent across the Case 2–Case 3 boundary.

B. Results for the Nominal Game

  • Under condition (11), Player 1 never benefits from giving resources, and Case 1 has no mutually beneficial transfer.Source: B. Results for the Nominal Game · p. 7
  • The Case 2 and Case 3 conditions both express that Player 2’s payoff increases when they begin giving resources, and they are equivalent on the boundary between those cases.The condition changes form but not truth value when a game crosses from Case 2 into Case 3, allowing analysis of uncertainty spanning both cases.Source: B. Results for the Nominal Game · p. 7
  • A transfer that strictly increases the giver’s payoff also strictly increases the receiver’s payoff.Source: B. Results for the Nominal Game · p. 7
  • When Player 2 gives and gains, Player 1’s payoff increases along the smaller transfers considered across Cases 1, 2, and 3.The proof uses increasing payoff with Player 1’s budget in Case 1, the Case 2 payoff’s dependence on the players’ budgets, and a positive derivative in Case 3.Source: B. Results for the Nominal Game · p. 7

C. Proofs for Adversary Uncertainty

For adversary-budget uncertainty, robustness is determined by whether every budget consistent with the uncertainty permits a mutually beneficial transfer. The proof reduces this to avoiding Case 1 and meeting the applicable Case 2 or Case 3 condition across the possible budgets.

C. Proofs for Adversary Uncertainty

  • A transfer is robust to adversary-budget uncertainty exactly when every possible adversary budget yields a game with a mutually beneficial transfer.The proof uses continuity of marginal payoff to show that if each such game admits a transfer, a sufficiently small transfer works across the full budget interval.Source: C. Proofs for Adversary Uncertainty · pp. 7–8
  • Every possible adversary budget must avoid Case 1, which is equivalent to δ < α1(G).Avoiding Case 1 is hardest for the weakest possible adversary budget.Source: C. Proofs for Adversary Uncertainty · p. 8
  • In Case 2, the condition becomes harder as the adversary’s budget grows and becomes tight at A ≜ X_A(1 + α2(G)); the corresponding criterion is exactly δ < α2(G).Source: C. Proofs for Adversary Uncertainty · p. 8
  • The Case 3 condition is independent of the adversary’s budget, and it agrees with the Case 2 condition at their boundary.Source: C. Proofs for Adversary Uncertainty · p. 8

D. Proofs for Ally Uncertainty

The proof characterizes robust transfers under ally uncertainty by restricting the transfer direction and identifying the budget conditions that must hold across Player 1’s uncertainty set. The characterization divides games according to whether X_A exceeds β5(G)X2.

Lemma 9

  • No positive transfer is δ-robust to ally uncertainty; for a negative transfer, robustness requires Player 2 to benefit in every game in GX1(G, δ).Player 1 cannot be the donor, and δ < β1(G) is necessary for Player 2 to benefit even when Player 1’s budget is largest in the uncertainty set.Source: D. Proofs for Ally Uncertainty · p. 8

Proof of Theorem 5

  • For a negative transfer, Player 1’s payoff condition holds automatically, so robustness reduces to Player 2 benefiting throughout GX1(G, δ).The proof then checks the largest possible Player 1 budget: if a mutually beneficial transfer exists there, Lemma 10 implies one exists at every smaller budget.Source: D. Proofs for Ally Uncertainty · p. 8
  • Given δ < β1(G), a robust transfer exists iff G at Xmax = X1(1 + δ) has a mutually beneficial transfer; continuity then makes one sufficiently small τ < 0 work across all possible budgets.Source: D. Proofs for Ally Uncertainty · p. 8

Theorem 5

  • If X_A ≤ β5(G)X2, a robust transfer exists exactly when δ < β1(G), δ < β2(G), and δ < β4(G); if X_A > β5(G)X2, exactly when δ < β6(G).The first condition combines the necessary bound δ < β1(G) with the thresholds for avoiding Case 1 and satisfying the Case 2 condition.Source: D. Proofs for Ally Uncertainty · p. 9
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