Stake-governed tug-of-war and the biased infinity Laplacian

TL;DR

The paper introduces stake-governed tug-of-war on graphs, focusing on allocated-budget play on root-reward trees and a leisurely move rule. It establishes that, under leisure and tree-structure assumptions, the game value aligns with the $\lambda$-biased infinity-harmonic function $h(\lambda,\cdot)$ and identifies a precise stake formula linking spatial gradients of $h$ to the remaining budgets. A central contribution is the explicit stake function $\text{Stake}(\varepsilon,\lambda,v)$, derived from a perturbative saddle-point analysis and interpreted as a ratio of short-term gain to long-term cost; the work also provides a detailed decomposition (Peres-Šunić/MS) enabling explicit computation of $h(\lambda,\cdot)$ on root-reward trees. Finite-horizon analyses and Poisson-approximation arguments are developed to justify the infinite-horizon results, with extensive exploration of normal-form contours on simple graphs to illustrate when a global saddle exists. The study forges a bridge between allocated-budget game theory and PDE-type harmonic analysis, offering explicit algorithms and posing open questions for broader graph classes, continuous limits, and applications to resource allocation and political advertising.

Abstract

In tug-of-war, two players compete by moving a counter along edges of a graph, each winning the right to move at a given turn according to the flip of a possibly biased coin. The game ends when the counter reaches the boundary, a fixed subset of the vertices, at which point one player pays the other an amount determined by the boundary vertex. Economists and mathematicians have independently studied tug-of-war for many years, focussing respectively on resource-allocation forms of the game, in which players iteratively spend precious budgets in an effort to influence the bias of the coins that determine the turn victors; and on PDE arising in fine mesh limits of the constant-bias game in a Euclidean setting. In this article, we offer a mathematical treatment of a class of tug-of-war games with allocated budgets: each player is initially given a fixed budget which she draws on throughout the game to offer a stake at the start of each turn, and her probability of winning the turn is the ratio of her stake and the sum of the two stakes. We consider the game played on a tree, with boundary being the set of leaves, and the payment function being the indicator of a single distinguished leaf. We find the game value and the essentially unique Nash equilibrium of a leisurely version of the game, in which the move at any given turn is cancelled with constant probability after stakes have been placed. We show that the ratio of the players' remaining budgets is maintained at its initial value $λ$; game value is a biased infinity harmonic function; and the proportion of remaining budget that players stake at a given turn is given in terms of the spatial gradient and the $λ$-derivative of game value. We also indicate examples in which the solution takes a different form in the non-leisurely game.

Figures (4)

• Figure 1: Normal form contour plots of $[0,\lambda] \times [0,1] \longrightarrow [0,1]: (a,b) \mapsto {\rm Val}(1,\lambda,v,a,b)$ for vertex $v = 2$ in the line graph $(L_3,\sim,{\bf 1}_3)$, drawn under the assumptions [A] of the GloshConch. The $\lambda$-values are $1/2$, $1$ and $3$ from left to right.
• Figure 2: The $T$ graph and five normal form contour plots of $[0,\lambda] \times [0,1] \longrightarrow [0,2]: (a,b) \mapsto {\rm Val}(1,\lambda,N,a,b)$, where $N$ is the north vertex and the GloshConch's assumptons are in force. The values of $\lambda$ in the upper middle and right are $0.9$ and $1$; on the lower left, middle and right, they are $\lambda_c - 1/10$, $\lambda_c$ and $\lambda_c + 1/10$.
• Figure 3: Plotting the Poisson game: normal form contour plots of $[0,\infty)^2 \longrightarrow \mathbb{R}: (a,b) \mapsto \Phi(\lambda,N,a,b)$ for the $T$ graph. From left to right, the $\lambda$-values are $3/2$, $\lambda_c = 1.618 \cdots$, and $2$. The left plot equally depicts $\Phi(3/2,2,a,b)$ for the line graph $L_3$, and the right one depicts $\Phi(2,1,a,b)$ for $L_2$. The global minimax point $(\lambda b_0,b_0)$, with $b_0$ specified in (\ref{['e.bzero']}), is marked with a yellow cross in the left and right plots. The middle plot depicts the minimum of $\Phi(\lambda_c,2,a,b)$ for $L_3$ and $\Phi(\lambda_c,1,a,b)$ for $L_2$, and the two yellow crosses mark the predicted saddle point $(\lambda b_0,b_0)$ for these two functions. The minimum reflects Mina's capacity to decide between nominating vertex $S$ or vertex $1$.
• Figure 4: The right root-reward tree is the journey-data tree of the left one. Each tree has depth three; the depth-zero mainlines are emboldened, the depth-one mainlines are dashed, the depth-two and three mainlines are densely and sparsely dotted, respectively. The junctions of vertex $v$ on the left are indicated. The vertices $v$ and $\Phi(v)$ have journey data $\{ (3,1),(4,2),(3,1) \}$ in their respective trees. For Maxine to win from $v$ on the left, she must triumph in three rounds: by reaching $j_2$, $j_1$ and $j_0$. For $\lambda \in (0,\infty)$, the $\lambda^{-1}$-slope of the path in the MS decomposition on which the counter is found rises with each round at which Maxine succeeds: on root-reward trees, tug-of-war for Maxine resembles a gameshow where a wrong answer means the contestant's exit, and the pitch of whose music rises by a semitone with each passing round.

Theorems & Definitions (62)

• Definition 1.1
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• Definition 1.3
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• Theorem 1.8
• Proposition 1.9
• Definition 1.10