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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.09988 |
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| _version_ | 1866914551879958528 |
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| author | Lee, Duan-Shin |
| author_facet | Lee, Duan-Shin |
| contents | In this paper we propose a Bayesian game to allocate resources.
In this game, there are $c$ units of resources to be
allocated to $n$ players. Agent $i$ has a demand of $V_i$ units of resources
and takes action $X_i$ according to a strategy function $s_i$, \ie $X_i=s_i(V_i)$.
Payoffs are setup such that player $i$ is contented with no more than $V_i$
units of resources. We assume that resources are granted to the players
on a smallest-request-first and all-or-nothing basis.
For this game with two players, we analyze the equilibrium strategy
functions mathematically within the family of alternating identity-and-flat (AIF)
functions. We show that Nash equilibrium profiles consist of two identity functions,
two AIF functions with a common switch point, or two AIF functions with one
and three switch points, respectively.
For an $n$-player game with a large $n$ and a large $c_n$
of order $O(n)$, we present a mean-field first order approximation and a
second-order Gaussian approximation for its
equilibrium strategy function. The first-order analysis obtains an equilibrium
AIF function with one switch point. In Gaussian analysis of large games, we
propose a construction algorithm. This construction algorithm begins in searching
within the family of AIF functions. If a gradient conflict condition occurs, the game
enters a chattering regime, in which players
play a continuous, strictly increasing strategy function that is not an identity nor a flat function.
Conceptually one can view the chattering regime as if players alternate between
a slope-one strategy and a flat strategy infinitely fast in order to sustain a high payoff.
We prove that the construction algorithm always
obtains a Nash equilibrium and terminates in a finite number of steps.
We present several numerical examples for the two player game as well as the Gaussian model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09988 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Resource Allocation Game and its Equilibrium Strategies Lee, Duan-Shin Computer Science and Game Theory In this paper we propose a Bayesian game to allocate resources. In this game, there are $c$ units of resources to be allocated to $n$ players. Agent $i$ has a demand of $V_i$ units of resources and takes action $X_i$ according to a strategy function $s_i$, \ie $X_i=s_i(V_i)$. Payoffs are setup such that player $i$ is contented with no more than $V_i$ units of resources. We assume that resources are granted to the players on a smallest-request-first and all-or-nothing basis. For this game with two players, we analyze the equilibrium strategy functions mathematically within the family of alternating identity-and-flat (AIF) functions. We show that Nash equilibrium profiles consist of two identity functions, two AIF functions with a common switch point, or two AIF functions with one and three switch points, respectively. For an $n$-player game with a large $n$ and a large $c_n$ of order $O(n)$, we present a mean-field first order approximation and a second-order Gaussian approximation for its equilibrium strategy function. The first-order analysis obtains an equilibrium AIF function with one switch point. In Gaussian analysis of large games, we propose a construction algorithm. This construction algorithm begins in searching within the family of AIF functions. If a gradient conflict condition occurs, the game enters a chattering regime, in which players play a continuous, strictly increasing strategy function that is not an identity nor a flat function. Conceptually one can view the chattering regime as if players alternate between a slope-one strategy and a flat strategy infinitely fast in order to sustain a high payoff. We prove that the construction algorithm always obtains a Nash equilibrium and terminates in a finite number of steps. We present several numerical examples for the two player game as well as the Gaussian model. |
| title | A Resource Allocation Game and its Equilibrium Strategies |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2605.09988 |