Potential game

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In
game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
, a game is said to be a potential game if the incentive of all players to change their
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can be expressed using a single global function called the potential function. The concept originated in a 1996 paper by Dov Monderer and
Lloyd Shapley Lloyd Stowell Shapley (; June 2, 1923 – March 12, 2016) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
. The properties of several types of potential games have since been studied. Games can be either ''ordinal'' or ''cardinal'' potential games. In cardinal games, the difference in individual payoffs for each player from individually changing one's strategy, other things equal, has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same. The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure
Nash equilibria In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is the most common way to define the solution concept, solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each playe ...
can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function. Potential games can be studied as
repeated games In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game (called a stage game). The stage game is usually one of the well-studied list of games in game theory, 2-person games. Repeated gam ...
with state so that every round played has a direct consequence on game's state in the next round. Marden, J., (2012) State based potential games http://ecee.colorado.edu/marden/files/state-based-games.pdf This approach has applications in distributed control such as distributed resource allocation, where players without a central correlation mechanism can cooperate to achieve a globally optimal resource distribution.

# Definition

We will define some notation required for the definition. Let $N$ be the number of players, $A$ the set of action profiles over the action sets $A_$ of each player and $u$ be the payoff function. A game $G=\left(N,A=A_\times\ldots\times A_, u: A \rightarrow \reals^N\right)$ is: * an exact potential game if there is a function $\Phi: A \rightarrow \reals$ such that $\forall i, \forall ,\ \forall$, ::$\Phi\left(a\text{'}_,a_\right)-\Phi\left(a\text{'}\text{'}_,a_\right) = u_\left(a\text{'}_,a_\right)-u_\left(a\text{'}\text{'}_,a_\right)$ ::That is: when player $i$ switches from action $a\text{'}$ to action $a\text{'}\text{'}$, the change in the potential equals the change in the utility of that player. * a weighted potential game if there is a function $\Phi: A \rightarrow \reals$ and a vector $w \in \reals_^N$ such that $\forall i,\forall ,\ \forall$, ::$\Phi\left(a\text{'}_,a_\right)-\Phi\left(a\text{'}\text{'}_,a_\right) = w_\left(u_\left(a\text{'}_,a_\right)-u_\left(a\text{'}\text{'}_,a_\right)\right)$ * an ordinal potential game if there is a function $\Phi: A \rightarrow \reals$ such that $\forall i,\forall ,\ \forall$, ::$u_\left(a\text{'}_,a_\right)-u_\left(a\text{'}\text{'}_,a_\right)>0 \Leftrightarrow \Phi\left(a\text{'}_,a_\right)-\Phi\left(a\text{'}\text{'}_,a_\right)>0$ * a generalized ordinal potential game if there is a function $\Phi: A \rightarrow \reals$ such that $\forall i, \forall ,\ \forall$, ::$u_\left(a\text{'}_,a_\right)-u_\left(a\text{'}\text{'}_,a_\right)>0 \Rightarrow \Phi\left(a\text{'}_,a_\right)-\Phi\left(a\text{'}\text{'}_,a_\right) >0$ *a best-response potential game if there is a function $\Phi: A \rightarrow \reals$ such that $\forall i\in N,\ \forall$, ::$b_i\left(a_\right)=\arg\max_ \Phi\left(a_i,a_\right)$ where $b_i\left(a_\right)$ is the best action for player $i$ given $a_$.

# A simple example

In ''a'' 2-player, 2-action game with externalities, individual players' payoffs are given by the function , where is players i's action, is the opponent's action, and ''w'' is ''a'' positive
externality In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plan ...

from choosing the same action. The action choices are +1 and −1, as seen in the
payoff matrix In game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a un ...
in Figure 1. This game has ''a'' potential function . If player 1 moves from −1 to +1, the payoff difference is . The change in potential is . The solution for player 2 is equivalent. Using numerical values , , , this example transforms into ''a'' simple battle of the sexes, as shown in Figure 2. The game has two pure Nash equilibria, and . These are also the local maxima of the potential function (Figure 3). The only
stochastically stable equilibrium In game theory, a stochastically stable equilibrium is a solution concept, refinement of the evolutionarily stable state in evolutionary game theory, proposed by Dean Foster and Peyton Young. An evolutionary stable state S is also stochastically st ...
is , the global maximum of the potential function. A 2-player, 2-action game cannot be ''a'' potential game unless :