Samaritan's Dilemma: Classical and quantum strategies in Welfare Game
S Kaya, Junichi Shimamura, Fumiaki Morikoshi +2 more
Effects of classical/quantum correlations and operations in game theory are analyzed using Samar- itan's Dilemma. We observe that introducing either quantum or classical correlations to the game results in the emergence of a unique or multiple Nash equilibria (NE) which do not exist in the original classical game. It is shown that the strategies creating the NE and the amount of payoffs the players receive at these NE's depend on the type of the correlation. We also discuss whether the Samaritan can resolve the dilemma acting unilaterally. PACS numbers: 03.67.-a, 02.50.Le (α, β) where α ∈ SA and β ∈ SB can be written as uA(α, β). In classical game theory, any game is fully described by its payoff matrix. Based on the nature of payoffs, games can be classified in three different ways (i) symmetric, uA(α, β) = uB(β, α), and asymmetric games, uA(α, β) 6 uB(β, α), (ii) zero-sum, uA(α, β)+uB(α, β) = 0 for ∀α ∈ SA, and non-zero sum games, if for ∃α ∈ SA and ∃β ∈ SB, uA(α, β) + uB(α, β) 6 0 , and (iii) coor- dination if the game has at least one Nash equilibrium (NE), and discoordination games if there is no NE in pure strategies (15). Nash equilibrium is the most commonly used equilib- rium concept for strategic games, and it can be regarded as the steady state of the strategic interaction. The con- cept of NE is based on the premises that each player acts rationally according to the belief he/she has on the other player, and that the beliefs of each player about the other one is correct. Once the players acts according to NE, no one can take another action to unilaterally deviate from it in order to increase his/her payoff. In an NE, each player's choice of action is the best response to the ac- tions taken by the other player. Although in pure strate- gies an NE need not exist, there is always at least one NE in mixed strategies where the players are allowed to ran- domize among their action. The most common difficulty encountered in the concept of NE is that NE need not be unique. There might be multiple NE's which avoids making sharp decisions. In such cases, certain NE's can be isolated as focal in that they are clearly better for all players, and they yield a higher payoff to every player than any other NE's. If there exists only one NE like this, then players will be self-enforced to play it, thus solving
•Nov 12, 2003•2 citations