Fortune's algorithm

2013-11-27T13:27:45Z

131.155.204.198: Removed a line that was outdated in Pseudocode.

In [[game theory]], a game is said to be a '''potential game''' if the incentive of all players to change their [[strategy (game theory)|strategy]] can be expressed using a single global function called the '''potential function'''. The concept was proposed in 1973 by [[Robert W. Rosenthal]].

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 [[payoff]]s for each player from individually changing one's strategy ''[[ceteris paribus]]'' 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 equilibrium|Nash equilibria]] 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.

==Definition==
We will define some notation required for the definition. Let <math>N</math> be the number of players, <math>A</math> the set of action profiles over the action sets <math>A_{i}</math> of each player and <math>u</math> be the payoff function.

A game <math>G=(N,A=A_{1}\times\ldots\times A_{N}, u: A \rightarrow \reals^N) </math> is:

* an '''exact potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})</math>

::That is: when player <math>i</math> switches from action <math>a'</math> to action <math>a''</math>, the change in the potential equals the change in the utility of that player.

* a '''weighted potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> and a vector <math>w \in \reals_{++}^N</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = w_{i}(u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i}))</math>

* an '''ordinal potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Leftrightarrow
\Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i})>0</math>

* a '''generalized ordinal potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Rightarrow
\Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) >0 </math>

*a '''best-response potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math>\forall i\in N,\ \forall {a_{-i}\in A_{-i}}</math>,
::<math>b_i(a_{-i})=\arg\max_{a_i\in A_i} \Phi(a_i,a_{-i})</math>
where <math>b_i(a_{-i})</math> is the best payoff for player <math>i</math> given <math>a_{-i}</math>.

==A simple example==
{{Payoff matrix | Name = Fig. 1: Potential game example
| 2L = +1 | 2R = –1 |
1U = +1 | UL = {{nowrap|+b1+w, +b2+w}} | UR = {{nowrap|+b1–w, –b2–w}} |
1D = –1 | DL = {{nowrap|–b1–w, +b2–w}} | DR = {{nowrap|–b1+w, –b2+w}} }}
In a 2-player, 2-strategy game with externalities, individual players' payoffs are given by the function {{nowrap|ui(si, sj)}} = {{nowrap|bi si + w si sj}}, where si is players i's strategy, {{nowrap|sj}} is the opponent's strategy, and w is a positive [[externality]] from choosing the same strategy. The strategy choices are +1 and −1, as seen in the [[payoff matrix]] in Figure 1.

This game has a potential function {{nowrap|P(s1, s2)}} = {{nowrap|b1 s1 + b2 s2 + w s1 s2}}.

If player 1 moves from −1 to +1, the payoff difference is Δu1 = {{nowrap|u1(+1, s2) – u1(–1, s2)}} = {{nowrap|2 b1 + 2 w s2}}.

The change in potential is ΔP = {{nowrap|P(+1, s2) – P(–1, s2)}} = {{nowrap|(b1 + b2 s2 + w s2) – (–b1 + b2 s2 – w s2)}} = {{nowrap|2 b1 + 2 w s2}} = Δu1.

The solution for player 2 is equivalent. Using numerical values b1 = 2, b2 = −1, w = 3, this example transforms into a simple [[battle of the sexes (game theory)|battle of the sexes]], as shown in Figure 2. The game has two pure Nash equilibria, (+1, +1) and (−1, −1). These are also the local maxima of the potential function (Figure 3). The only [[stochastically stable equilibrium]] is (+1, +1), the global maximum of the potential function.
<center>
{|
|-
| width=50% |
{{Payoff matrix | Name = Fig. 2: Battle of the sexes (payoffs)
| 2L = +1 | 2R = –1 |
1U = +1 | UL = 5, 2 | UR = –1, –2 |
1D = –1 | DL = –5, –4 | DR = 1, 4 }}
| width=50% |
{{Payoff matrix | Name = Fig. 3: Battle of the sexes (potentials)
| 2L = +1 | 2R = –1 |
1U = +1 | UL = '''4''' | UR = '''0''' |
1D = –1 | DL = '''–6''' | DR = '''2''' }}
|}
</center>
A 2-player, 2-strategy game cannot be a potential game unless
:<math>
[u_{1}(+1,-1)+u_1(-1,+1)]-[u_1(+1,+1)+u_1(-1,-1)] =
[u_{2}(+1,-1)+u_2(-1,+1)]-[u_2(+1,+1)+u_2(-1,-1)]
</math>

==References==
* Dov Monderer and Lloyd S. Shapley: "Potential Games", ''Games and Economic Behavior'' 14, pp. 124–143 (1996).
* Emile Aarts and Jan Korst: ''Simulated Annealing and Boltzmann Machines'', John Wiley & Sons (1989) ISBN 0-471-92146-7

==External links==
* Lecture notes of Yishay Mansour about [http://www.math.tau.ac.il/~mansour/course_games/scribe/lecture6.pdf Potential and congestion games]

[[Category:Game theory]]

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Fortune's algorithm