Structure of ε-Nash Equilibrium in Graphical Games
Ruiqi (Sally) Li, Class of 2018
Game theory provides a mathematical framework to model and study human behavior in strategic
scenarios [1]. In our research, we focus on the predominant branch of game theory known as noncooperative games. In 1950, John Nash proved that every finite game has at least one mixed-strategy
Nash equilibrium, which is the standard solution concept in non-cooperative games. Decades later, it was
proved that the computation of a mixed-strategy Nash equilibrium is “PPAD-hard” (or computationally
intractable under usual assumptions). Due to this difficulty in computation, we decided to explore mixedstrategy ε-Nash equilibrium where ε signifies the notion of approximation. In an ε-Nash equilibrium a
player’s payoff can be at most ε worse off compared to the optimal payoff, and ε = 0 signifies a true Nash
equilibrium. The major question we have pursued is: Is there any correlation between computationally
hard games (i.e., games where the players have conflicting preferences) and their equilibrium structure?
We started the research by implementing an algorithm [2] that computes the set of all ε-Nash
equilibria for a tree-structured graphical game. Here, a graphical game is a special class of games where
an underlying network connects the players and a player's payoff directly depends on her action and the
actions of her neighbors in that network. In a tree-structured graphical game, the underlying network is a
tree, that is, a connected network with no cycle. We chose this specific class of games as a first step
because the algorithm can be easily generalized to cyclic games later on. We decided to use Python
programming language because there already exist many useful packages for network operations that are
convenient to use. The inputs to the program are the value of desired ε, the grid sizes for discretizing the
probability and payoff spaces. Our program first generates a set of random tree-structured graphical
games and stores each in a separate file. We decided on a tree-structured network with seven nodes,
where each internal node has two children. This gave us a small balanced tree for which the game was not
too time-consuming to solve. We also decided to use 0.05 and 0.10 as our values of interest for ε and 41
for both the probability and payoff grid sizes. Our program uses the algorithm given by Ortiz and Irfan [2]
to compute the set of all ε-Nash equilibria for each game. It also systematically stores these equilibria.
We did two things to understand the output. First, we utilized the multidimensional scaling tool
(XLSTAT by Microsoft) to visualize the set of all ε-Nash equilibria in a 2D plot to see if there exists any
structure. Second, we wrote a C++ program to condense the ε-Nash equilibria information across different
games so that we can easily compare the running time for each game and the structure of the equilibria.
Regarding our first question of whether there is a correlation between computation time and the structure
of the equilibra, we realized that for the “easier” games, the grid size of 41 was not large enough, because
essentially every single point on this coarse discretized grid was an ε-Nash equilibrium. For the games
where 41 is a large enough grid size, we tend to see a clear line or a set-of-parallel-lines structure in the
visualization of the equilibria. This structure indicates that the equilibria are close to one another and once
one is computed, it takes a lot less time to find another. In future, we would like to extend this study to
answer whether there is any connection between the structure of a game (i.e., tree vs. cyclic graphs) and
its set of ε-Nash equilibria.
References
[1] Levine, David K. "Economic and Game Theory: What Is Game Theory?" University of California,
Los Angeles, 2016. (http://levine.sscnet.ucla.edu/general/whatis.htm).
[2] Ortiz, Luis E., and Irfan, Mohammad T. "FPTAS for Mixed-Strategy Nash Equilibria in Tree
Graphical Games and Their Generalizations." arXiv preprint arXiv:1602.05237, 2016.
Faculty Mentor: Professor Mahammad T. Irfan
Funded by: Kibbe Summer Research Program