Abstract
An algorithm is presented in this note for determining all Berge equilibria for an n-person game in normal form. This algorithm is based on the notion of disappointment, with the payoff matrix (PM) being transformed into a disappointment matrix (DM). The DM has the property that a pure strategy profile of the PM is a BE if and only if (0,…,0) is the corresponding entry of the DM. Furthermore, any (0,…,0) entry of the DM is also a more restrictive Berge-Vaisman equilibrium if and only if each player’s BE payoff is at least as large as the player’s maximin security level.
1. Introduction
In a Berge equilibrium (BE) for an -person game, every player has pure strategies that maximized the remaining player’s payoff. The BE was intuitively defined for pure strategies in [1] as a refinement to the Nash equilibrium (NE) [2]. The BE was formalized in [3] as a game-theoretic solution concept modeling mutual support and cooperation, as opposed to the selfishness of the NE. However, the BE was not studied extensively until recently. For example, see [4–9]. Only [6, 7] offer approaches for obtaining BEs, though, and these methods are stated abstractly without examples.
A simple algorithm is presented here for computing all BEs for an -person game in normal form, with the set of players, the finite set of pure strategies for player , and the von Neumann-Morgenstern utility of player for a pure strategy profile . The BE is defined as follows, where an incomplete strategy profile denotes a member of .
Definition 1. The strategy profile is a BE of if and only if The BE is further called a Berge-Vaisman equilibrium (BVE) if and only if the maximin security levels of the players satisfy so that no player is guaranteed better payoff than that given by .
In Section 2 we define the disappointment matrix (DM) corresponding to the payoff matrix (PM) for and prove that a pure strategy is a BE if and only if is the corresponding entry of the DM. In Section 3 the approach of Section 2 is formalized as an algorithm to obtain all BEs and BVEs for . In Section 4 a computational example is presented for .
2. The Disappointment Matrix
The disappointment function is defined as a transformation of a player’s payoffs to losses.
Definition 2. For the game , the disappointment incurred by any player choosing and the other players choosing is defined as
The disappointment incurred by player for a strategy profile is thus the difference between the best payoff that player could obtain by choosing and the actual payoff that player would obtain for the strategy profile . This definition immediately gives the following result.
Theorem 3. The pure strategy profile is a BE for the game if and only if the disappointment for all . Moreover, a BE is also a BVE if and only if for all .
Proof. The strategy profile is a BE if and only if it satisfies (1) of Definition 1. However, (1) holds if and only if in (3) for all . To complete the proof, the BE is also a BVE if and only if it satisfies (2).
The DM of a game is now defined.
Definition 4. The disappointment matrix of is the matrix obtained from its PM by replacing by for all .
3. Algorithm
Consider the game in which each player has pure strategies. Denote the th strategy of player by . Algorithm 1 describes the algorithm BECOMP for obtaining all BEs and BVEs for from its DM. The computational complexity of BECOMP is because of the unavoidable enumeration in Step 4 of all strategy profiles , for each player .
| ||||||||||||||||||
4. Example
For we use the notation of BECOMP with and consider the game with a PM of Table 1. The DM for is obtained from the PM via BECOMP and shown in Table 2. The unique BE for is with associated payoffs . However, is not a BVE since . Player 1 is guaranteed better payoff by choosing instead of .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.