🤖 AI Summary
This study addresses the fair and efficient allocation of limited resources among multiple agents under information asymmetry, analyzing strategic equilibria in such settings. A Bayesian game model is formulated wherein agents submit requests based on private demand valuations, and resources are allocated according to a “smallest-request-first, all-or-nothing” rule. The work provides the first systematic characterization of equilibrium structures in two-player games under alternating identity-flat (AIF) strategies, rigorously establishing three distinct forms of Nash equilibria. For large-scale settings, the paper introduces mean-field first-order and Gaussian second-order approximation methods and devises a finite-step convergent algorithm. Numerical experiments validate the theoretical findings and uncover a novel equilibrium behavior termed the “jittering mechanism.”
📝 Abstract
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.