On the exact polynomial time algorithm for a special class of bimatrix game

The Strategy Elimination (SE) algorithm was proposed in [2] and implemented by a sequence of Linear Programming (LP) problems. In this paper an efficient explicit solution is developed and the convergence to the Nash Equilibrium is proven.


Introduction
This paper investigates an example of quadratic bimatrix game model that describes the competition between inspectors and violators. We call them Inspector Games (IG). The IG is similar but not identical to the well known Inspection Game [3]. To solve large scale game problems and to prepare examples of game theory studies, it is essential to use polynomial time algorithms. No polynomial time algorithm is known for obtaining Nash Equilibrium (NE) of bimatrix games in general [4]. Therefore, an important task is to define a subset BG problems where NE can be obtained in polynomial time. In the paper this is done for Inspector Game.

Profit functions
The payoff of the first player is expressed by a matrix u(i, j) where i = 1, . . . , m denote moves (pure strategies) of the first player and j = 1, . . . , n represent moves of the second. The payoff of the second player is v(i, j). The expected profit U and V of the first and second players are defined as expected values of payoffs u(i, j) and v(i, j) Here x = (x 1 , . . . , x m ) and y = (y 1 , . . . , y n ) are probabilities (mixed strategies) of moves i and j.

Equilibrium testing in the quadratic bimatrix game
The equilibrium can be tested directly by the conditions of Nash equilibrium. First define the "contract" profits U * , V * assuming that both players keep the contract solution (x * , y * ) Then define the maximal profits U max , V max of players assuming that opposite players keep the contract solution (x * , y * ) The contract profits U * , V * are compared with the values U max , V max , and if the contract solution x * i > 0, y * j > 0 is recorded as an equilibrium strategy.

Irrelevant Fraud model
The Irrelevant Fraud (IF) model defines conditions where no incentive to break the contract solution (x * , y * ) exists.
In this paper, an explicit solution is presented. If the solution happens to be positive, meaning that x i > 0, y j > 0, i, j = 1, . . . , m, than according to the conditions of [2] that is NE. Variables x i and y j can be zero or negative, too. Then the quadratic elimination (QE) algorithm [2] can be used which provides NE in a specific bimatrix game called as the Inspector Game.

Profit functions
Denote by x = (x 1 , . . . , x m ), x i 0, i x i = 1 the inspection vector and by y = (y 1 , . . . , y m ), y j 0, j y j = 1 the violation vector. Here x i denotes the inspection probability of the object i and y j means the violation probability of the object j. Denote by u(i, j) the inspector's payoff when the object i is inspected and the object j is violated. Denote by v(i, j) the violator's payoff when the object i is inspected and the object j is violated. Functions U (x, y) and V (x, y) denote the inspector's and violator's profit functions using inspection and violation vectors x, y. These vectors define probabilities of inspection and violation. Payoffs of the Inspector Game (IG) and Here p i ∈ (0, 1] is the probability of detecting the violation if it happens in the object i, q i ∈ (0, 1] is the probability of the violation in the object i, and g i ∈ (0, ∞) is the payoff (potential) of the violation in the object i.
The profit functions at given inspection and violation vectors x and y Here u(i, j) = v(i, j), thus the inspection model is a bimatrix game [1].

Explicit solution of the irrelevant Fraud model
It follows from (1)

Strategy elimination algorithm
The general idea of the Strategy Elimination Algorithm (SE) is to eliminate strategies that lead to non-positive IF solutions. First an IF solution for both players x, y is generated. If for some k, x k 0 or y k 0 and the solution of the Direct Search Algorithm (DS) [2] is not found, then both the strategies x k and y k are eliminated from the set of feasible strategies by setting them to zero x k = y k = 0.
Iterative application of this method generates a sequence of BG. The iteration stops if a positive IF solution of the reduced QBG is obtained, a DS solution is found, or just a single element remains. Thus no more than n such steps are needed.

Theorem 1. A pair of SE solutions (x, y) is a Nash equilibrium of Inspector Game.
Proof. After some iterations SE returns positive vectors x * , y * according to the conditions (1), (2) and (3) excluding inequalities. These vectors are calculated for reduced matrices u * (i, j) and v * (i, j). Denote by R = {r 1 , r 2 , . . . , r k }, k m a subset of a complete set M of inspection objects 1, 2, . . . , m. If the SE solution x * i , y * i , i ∈ R is positive then according to the conditions of [2] that is NE in subset R. From (6) follows that the general solution x = (x 1 , . . . , x m ), x j = 0, j ∈ M \ R satisfies inequality Then, from (7) we can write that That means that SE solution (x, y) is NE for all inspection objects.

Theorem 2. Complexity of the SE algorithm is of polynomial.
Proof. The SE algorithm reads the lists at most m−1 k=0 (m − k) times that means the polynomial time complexity.
The algorithm is implemented as an Java applet and is on the web: http:// optimum2.mii.lt/. The pseudo-code follows: NoSolutions ← true, n ← m 6.
for i ← 1 to n do 9. if . return x, y  Table 1 illustrates the efficiecy of explicit solution comparing to Linear Programming (LP) implementation used in [2]. The CPU time in seconds (by Intel(R), Core(TM)2, 2.4GHz, RAM 2GB, Linux, Fedora 8) concides with the data of the chart for the dimension m.

Web-based software
The software is implemented as an Java applet on the web http://optimum2.mii.lt/ and can be started by pointing "Start Bimatrix Game" in the section "Discrete optimization". By clicking the label "Generate" one generates a number of inspection objects. The Label "Calculate" starts NE search.

Concluding remarks
We need polynomial time algorithms to solve large scale game problems and to explore numerically examples in studies of game theory. No polynomial time algorithm obtaining Nash Equilibrium (NE) is known for Bimatrix Games (BG) in general. Therefore, an important task is to define a subset of BG problems where NE can be reached in polynomial time. In this paper an exact polynomial time algorithm is described for a special class of Bimatrix Game called as the Inspector Game.