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Zero-Determinant Strategies in Iterated Public Goods Game.

Pan L, Hao D, Rong Z, Zhou T - Sci Rep (2015)

Bottom Line: A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way.This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players.The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

View Article: PubMed Central - PubMed

Affiliation: CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China.

ABSTRACT
Recently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner's Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

No MeSH data available.


(a) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed in terms of the two linear inequalities in equations (18) and (19) in Supplementary Methods, except for the singular point (pC,N−1, pD,0) = (1, 0). (b) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed by the two linear inequalities in equations (24) and (25) in Supplementary Methods. The intersected region is a convex hull with four extreme points. This region shrinks as the gradients of the two confine lines approaches each other. (c) Log-log plot of the upper bound of r. The upper bound  is a monotonously decreasing function of the group size N, namely with the increasing of N, the allowed region of multiplication factor for an equalizer strategy shrinks.
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f3: (a) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed in terms of the two linear inequalities in equations (18) and (19) in Supplementary Methods, except for the singular point (pC,N−1, pD,0) = (1, 0). (b) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed by the two linear inequalities in equations (24) and (25) in Supplementary Methods. The intersected region is a convex hull with four extreme points. This region shrinks as the gradients of the two confine lines approaches each other. (c) Log-log plot of the upper bound of r. The upper bound is a monotonously decreasing function of the group size N, namely with the increasing of N, the allowed region of multiplication factor for an equalizer strategy shrinks.

Mentions: Moreover, according to equations (8) and (9), all the other 2N − 2 strategy components and the coefficients μ and ξ can be represented by pC,N−1 and pD,0. In Supplementary Methods, the monotonicity analysis affirms that as long as the probability constraints 0 ≤ pC,N−1 ≤ 1 and 0 ≤ pD,0 ≤ 1 are satisfied, the nontrivial equalizer strategies exist. Generally, the feasible regions of equalizer strategies are the intersections of two half-planes determined by pC,N−1 and pD,0, which can be obtained by linear programming. In Fig. 3, we illustrate the feasible regions of equalizer strategies under different cases of r and N, as well as the allowed upper bound of r versus different N. It is shown that as the increase of the number of player N, the allowed upper bound of r decreases with the number of players N, namely the feasible regions of equalizer strategies get narrow. Thus it is difficult for player 1 to pin his opponents’ payoff when more players participate in the game.


Zero-Determinant Strategies in Iterated Public Goods Game.

Pan L, Hao D, Rong Z, Zhou T - Sci Rep (2015)

(a) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed in terms of the two linear inequalities in equations (18) and (19) in Supplementary Methods, except for the singular point (pC,N−1, pD,0) = (1, 0). (b) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed by the two linear inequalities in equations (24) and (25) in Supplementary Methods. The intersected region is a convex hull with four extreme points. This region shrinks as the gradients of the two confine lines approaches each other. (c) Log-log plot of the upper bound of r. The upper bound  is a monotonously decreasing function of the group size N, namely with the increasing of N, the allowed region of multiplication factor for an equalizer strategy shrinks.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4543983&req=5

f3: (a) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed in terms of the two linear inequalities in equations (18) and (19) in Supplementary Methods, except for the singular point (pC,N−1, pD,0) = (1, 0). (b) The feasible region of the equalizer strategies when , which is determined by the intersection of the two half-planes formed by the two linear inequalities in equations (24) and (25) in Supplementary Methods. The intersected region is a convex hull with four extreme points. This region shrinks as the gradients of the two confine lines approaches each other. (c) Log-log plot of the upper bound of r. The upper bound is a monotonously decreasing function of the group size N, namely with the increasing of N, the allowed region of multiplication factor for an equalizer strategy shrinks.
Mentions: Moreover, according to equations (8) and (9), all the other 2N − 2 strategy components and the coefficients μ and ξ can be represented by pC,N−1 and pD,0. In Supplementary Methods, the monotonicity analysis affirms that as long as the probability constraints 0 ≤ pC,N−1 ≤ 1 and 0 ≤ pD,0 ≤ 1 are satisfied, the nontrivial equalizer strategies exist. Generally, the feasible regions of equalizer strategies are the intersections of two half-planes determined by pC,N−1 and pD,0, which can be obtained by linear programming. In Fig. 3, we illustrate the feasible regions of equalizer strategies under different cases of r and N, as well as the allowed upper bound of r versus different N. It is shown that as the increase of the number of player N, the allowed upper bound of r decreases with the number of players N, namely the feasible regions of equalizer strategies get narrow. Thus it is difficult for player 1 to pin his opponents’ payoff when more players participate in the game.

Bottom Line: A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way.This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players.The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

View Article: PubMed Central - PubMed

Affiliation: CompleX Lab, Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 611731, China.

ABSTRACT
Recently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner's Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents' payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

No MeSH data available.