Applying Risk Theory to Game Theory Tristan Barnett

Applying Risk Theory to Game Theory Tristan Barnett www. strategicgames. com. au strategicgames@hotmail. com Linearity Axiom Media Release The mathematical foundations of game theory as outlined in Von Neumann and Morgenstern (1944) is built around maximizing expectations. This is shown in a game theory axiom which states that numbers in the game matrix must be cardinal utilities and can be transformed by any positive linear function f(x)=ax+b, a>0 without changing the information they convey. Maximin P 1: A=0. 2, B=0. 8 Risk averse P 1: A=0, B=1 Game 1 A Fair Deal in Law is On the Cards Can the probability of success in cards be applied to disputes in law? Victoria University mathematician Dr Tristan Barnett believes it can. He has developed a formula that can be applied directly to dispute resolution to determine the value of arbitration between two parties in dispute. To develop his formula, Dr Barnett has drawn on his statistical research on profitable card games such as blackjack and video poker. Game 2 Player A may choose the risk averse strategy in Game 1 to eliminate the possibility of a -3 payout, but may choose the maximin strategy in Game 2 since is guaranteed at least a payout of +1. § This contradicts the Von Neumann-Morgernstern Linearity Axiom Nash Equilibrium Prisoner’s Dilemma By applying the same probability formula to a lawsuit that is used to maximize long-term profit in a casino game, Dr Barnett can determine whether it is worthwhile to file a lawsuit, given there are risks involved if unsuccessful in court. His research has recently been published in the journal Law, Probability & Risk. Arbitration in settling disputes is based around expectations, Dr Barnett said. “My research has revealed that the party attempting to recover payment will invariably receive an amount less than their initial expectation, ” he said. Game 3 Game 4 In Game 3, if P 1 plays strategy B, then P 1 is guaranteed a positive payout of at least 1 unit regardless of the strategies used by P 2. Whereas playing strategy A could give a player a payout of -1, even though 2 units is possible if both players cooperate. As long as strategy A for P 1 is less than 0. 5, then the expected payout will be positive regardless of the strategies used by P 2. § Based on this reasoning a solution to a game does not have to be Pareto optimal, which contradicts the Pareto Principle. In Game 4, if P 2 plays strategy B, then P 1 will always end up with a negative payout regardless of the strategy P 1 uses. If P 1 plays strategy B, then P 2 will always end up with a negative payout regardless of the strategy P 2 uses. Therefore cooperation is necessary in Game 4, in order for P 1 or both players to obtain a positive payout. Nash Arbitration “The party should accept this lower amount when it is offered, because of the possibility of losing if the dispute were to go to court. These findings contradict the current game theory framework as documented in established texts such as Game Theory and the Law. ” The Nash Equilibrium strategies are P 1: A=1/7, B=6/7 and P 2: A=2/3, B=1/3. The maximin strategies are P 1: A=4/9, B=5/9 and P 2: A=2/7, B=5/7. Playing the Nash Equilibrium mixed strategy of P 1: A=1/7, B=6/7 could result in a negative expected payout if P 2 plays strategy A. For this reason, the maximin strategy could also be identified as a No Regret Policy and by relaxing the linearity axiom: § Maximin strategies could also be considered as Nash Equilibrium strategies Dr Barnett’s arbitration formula arose from being part of a gambling syndicate, where the objective was to maximize long-term profit. His formula could be applied in arbitration to resolve disputes at bodies such as the Victorian Civil and Administrative Tribunal (VCAT). “This would provide a fairer outcome and potentially a more efficient process, thus easing the stress for both parties in dispute, ” Dr Barnett said. The general theory goes well beyond lawsuits, which has led to Dr Barnett presenting his findings at the Second Brazilian Workshop of the Game Theory Society in late July. Jim Buckell Senior Media Officer It could be argued that P 1 would be more inclined than P 2 to obtain an arbitration process rather than playing the game simultaneously against P 2 to obtain a payout. This is reflected by the observation that P 1 could end up with a negative payout by playing the game simultaneously against P 2, even though the expected payout for P 1 is guaranteed to be positive. Therefore it appears logical that a maximin strategy is used for P 1 to obtain the status quo for the Nash arbitration process. It also appears logical that the mixed Nash Equilibrium (assuming one exists) is used for P 2 to obtain the status quo for the Nash arbitration process.