Adaptive Evolution of Cooperation through Darwinian Dynamics in Public Goods Games

The linear or threshold Public Goods game (PGG) is extensively accepted as a paradigmatic model to approach the evolution of cooperation in social dilemmas. Here we explore the significant effect of nonlinearity of the structures of public goods on the evolution of cooperation within the well-mixed population by adopting Darwinian dynamics, which simultaneously consider the evolution of populations and strategies on a continuous adaptive landscape, and extend the concept of evolutionarily stable strategy (ESS) as a coalition of strategies that is both convergent-stable and resistant to invasion. Results show (i) that in the linear PGG contributing nothing is an ESS, which contradicts experimental data, (ii) that in the threshold PGG contributing the threshold value is a fragile ESS, which cannot resist the invasion of contributing nothing, and (iii) that there exists a robust ESS of contributing more than half in the sigmoid PGG if the return rate is relatively high. This work reveals the significant effect of the nonlinearity of the structures of public goods on the evolution of cooperation, and suggests that, compared with the linear or threshold PGG, the sigmoid PGG might be a more proper model for the evolution of cooperation within the well-mixed population

Action Being Character: A Promising Perspective on the Solution Concept of Game Theory

The inconsistency of predictions from solution concepts of conventional game theory with experimental observations is an enduring question. These solution concepts are based on the canonical rationality assumption that people are exclusively self-regarding utility maximizers. In this article, we think this assumption is problematic and, instead, assume that rational economic agents act as if they were maximizing their implicit utilities, which turns out to be a natural extension of the canonical rationality assumption. Implicit utility is defined by a player's character to reflect his personal weighting between cooperative, individualistic, and competitive social value orientations. The player who actually faces an implicit game chooses his strategy based on the common belief about the character distribution for a general player and the self-estimation of his own character, and he is not concerned about which strategies other players will choose and will never feel regret about his decision. It is shown by solving five paradigmatic games, the Dictator game, the Ultimatum game, the Prisoner's Dilemma game, the Public Goods game, and the Battle of the Sexes game, that the framework of implicit game and its corresponding solution concept, implicit equilibrium, based on this alternative assumption have potential for better explaining people's actual behaviors in social decision making situations

The Ultimatum Game.

<p>The Ultimatum Game.</p

The adaptive landscapes in the sigmoid PGG.

<p> is an ESS which sits at the top of the adaptive landscape. is an unstable minimum which sits at the bottom of the adaptive landscape. Parameters: , , , , , , , and ; ; and .</p

The Prisoner's Dilemma Game.

<p>The Prisoner's Dilemma Game.</p

The three kinds of structures of the PGG.

<p>(<b>Dash-dot</b>) The linear PGG, . (<b>Solid</b>) The sigmoid PGG, . (<b>Dashed</b>) The threshold PGG, if , and if .</p

An invasion simulation of Darwinian dynamics of the linear PGG.

<p>(<b>Upper-left</b>) Evolution of the frequencies of the ESS and the invader strategy starting from and respectively. (<b>Upper-right</b>) Evolution of the ESS and the invader strategy starting from and and ending up with the latter evolving to the former. (<b>Lower</b>) Evolution of the adaptive landscape and the two strategies: (i.e., before the invasion happens), is the global maximum; (i.e., the invasion happens), the adaptive landscape is elevated with still being the global maximum and being the global minimum; , the invader strategy climbs up with the adaptive landscape going down; , the invader strategy coincides with and reaches the top of the adaptive landscape, which falls back to the state before the invasion happens. Parameters: , , and .</p

The Public Goods Game.

<p>The Public Goods Game.</p

An invasion simulation of Darwinian dynamics in the sigmoid PGG.

<p>(<b>Upper-left</b>) Evolution of the frequencies of the ESS and the invader strategy starting from and respectively. (<b>Upper-right</b>) Evolution of the ESS and the invader strategy starting from and and ending up with the latter evolving to the former. (<b>Lower</b>) Evolution of the adaptive landscape and the two strategies: (i.e., before the invasion happens), is the global maximum; (i.e., the invasion happens), the adaptive landscape is reshaped with sitting at the left of the global maximum and being the global minimum; , the two strategies climb up so that the adaptive landscape is reshaped with the global maximum sitting between the two strategies; , the two strategies coincide and reach the top, at , of the adaptive landscape, which falls back to the state before the invasion happens. Parameters: , , and .</p

An invasion simulation of Darwinian dynamics in the threshold PGG which is approximated by where .

<p>(<b>Upper-left</b>) Evolution of the frequencies of the ESS and the invader strategy starting from and respectively. (<b>Upper-right</b>) Evolution of the ESS and the invader strategy starting from and and ending up with the former evolving to the latter. (<b>Lower</b>) Evolution of the adaptive landscape and the two strategies: (i.e., before the invasion happens), is the global maximum; (i.e., the invasion happens), the adaptive landscape is elevated with being the global minimum and being the local maximum; , the “ESS” climbs up towards the invader strategy with the latter keeping sitting at the local maximum of the reshaped adaptive landscape; , the “ESS” coincides with and reaches the top of the reshaped adaptive landscape, which means the success of the invader strategy and the failure of the “ESS”. Parameters: , , and .</p