1. Introduction

N-person games have been investigated by computer simulation under the assumption that the payoff functions are linear [1-4]. In real-life situations, however, the payoff functions are seldom linear [5]. For example, we have simulated a new game with parabolic payoff functions [6]. In this short note we will show the influence of their shape on the results of the simulations. The defectors’ payoff function will still be linear but parabolas will represent that of the cooperators. New papers about N-person games (for example, [7]) do not consider this problem at all. Agent-based simulation will be presented for the case when the agents are greedy simpletons who imitate the action of that of their neighbors who received the highest payoff for its previous action.

2. The Model

The individual agents may cooperate with each other for the collective interest or may defect, i. e., pursue their selfishinterests. Their decisions to cooperate or defect will accumulate over time to produce a result that will determine the success or failure of the given artificial society.

As usual, the payoff functions will be presented as functions of the number of cooperators related to the total number of agents (x). The payoff to each agent depends on its choice and on the distribution of other players among cooperators and defectors. We will assume that the payoff functions D(x) for the defectors are linear functions of this ratio x, but those for cooperators C(x) are quadratic. The two functions cross each other at a certain value of x_cross. The payoff function C(x) is the same for all cooperators and D(x) is the same for all defectors (uniform game).

We will use our agent-based model developed for simulated social and economic experiments with a large number of decision-makers operating in a stochastic environment [8]. The simulation environment is a two-dimensional array of the participating agents. Its size is limited only by the computer’s virtual memory. The behavior of a few million interacting agents can easily be observed on the computer’s screen.

There are two actions available to each agent, and each agent must choose between them (cooperation or defection). The cooperators and the defectors are initially distributed randomly over the array. In the iterative game the aggregate cooperation proportion changes in time, i.e., over subsequent iterations. At each iteration, every agent chooses an action according to the rewards to its neighbors. The software tool draws the array of agents in a window on the computer’s screen, with each agent in the array colored according to its most recent action. The updating occurs simultaneously for all agents.

After a certain number of iterations the proportion of cooperators stabilizes to either a constant value or oscillates around such a value. The experimenter can view and record the evolution of the society of agents as it changes in time. The outcome of the game strongly depends on the “personalities” of the agents, i.e. e. on the type of their responses to their environment. The software tool allows for a number of different personalities and their arbitrary combinations. In this work we assume that all agents are greedy simpletons.

3. Simulation

Throughout this work the total number of agents is 500x500=250,000, the initial ratio of cooperators is 50% and the neighborhood of each agent is one layer deep, i.e., each agent has exactly eight neighbors except those that are situated at the borders of the array.

We represent the payoff functions as

(1)

and

(2)

where a_1, b_1, c_1, and x_cross are the parameters of the experiments (the last is uniquely defined by the previous three).

We have performed computer simulation for hundreds of values of these parameters. The global ratio X(t) of the total number of cooperators in the entire array as a function of time (iterations) was observed for each choice of the parameter values. (Note that X is different from x that refers to an agent’s immediate neighbors only.) The final ratio of cooperators X_final around which X(t) oscillates represents the solution of the game. We present the results in three groups as follows.

We first fixed the values of b₁ = 1 and x_cross= 0.7. The only free parameter then is c_1. The payoff functions for several values of c₁ are shown in Figure 1. For comparison, a linear C(x) function is also presented in the figure.

The solution of the game X_final as a function of c₁ is given in Figure 2. As we see, the shape of the C(x) function has very little influence on the outcome of the game in this case.

Figure 1. Payoff (reward/penalty) functions for defectors (D) and cooperators (C). The horizontal axis (x) represents the ratio of the number of cooperators to the total number of agents in the neighborhood; the vertical axis is the payoff provided by the environment. In this case, D(x) = x and C(x) is defined by Equation (1) so that b1=1 and xcross=0.7. The only free parameter is c1. The C(x) function is shown for four different values of this parameter. For comparison, a linear C(x) function is also presented

Figure 2. The solution of the game Xfinal as a function of c1

Figure 3. Payoff functions for defectors (D) and cooperators (C). D(x) = x and C(x) is defined by Equation (1) so that c1=1 and xcross=0.7. The free parameter is now b1. The C(x) function is shown for four different values of this parameter. For comparison, a linear C(x) function is also presented

Figure 4. The solution of the game Xfinal as a function of b1

Figure 5. Refined solution: Xfinal around b1 = 0.57

Figure 6. Payoff functions for defectors (D) and cooperators (C). D(x) = x and C(x) is defined by Equation (1) so that b1=1 and c1=2, leaving xcross as the free parameter. The payoff functions for eight values of this parameter are shown

Next, let us fix the values of c₁ = 1 and x_cross= 0.7. The free parameter now is b_1. The payoff functions for several values of b₁ are shown in Figure 3. For comparison, a linear C(x) function is also presented in the figure.

The solution of the game X_final as a function of b₁ is given in Figure 4. We see a very sharp drop of X_final (from 0.88 to 0.64) around b₁ = 0.57. To investigate this effect in more detail, we refined the simulation in this region. The result is shown in Figure 5. As we see, the drop does not happen in one step, but a short sharp fluctuation takes place. We have observed a similar phenomenon in case of the N-person Chicken game [9].

Finally, we fixed the values of b₁ = 1 and c₁ = 2, leaving x_cross as the free parameter. The payoff functions for several values of this parameter are shown in Figure 6. The solution of the game X_final as a function of x_cross is given in Figure 7. As we can expect, the ratio of cooperators grows with the increase of x_cross because an increasing portion of the agents receive rewards. The increase is gradual until we reach x_cross = 0.7. Then X_final changes from 0.66 to 0.77. An even sharper jump happens at x_cross = 0.775 where X_final suddenly increases from 0.77 to 1.00 (full cooperation).

Figure 7. The solution of the game Xfinal as a function of xcross

4. Conclusions

Our simulations show that the shapes of the payoff functions have a profound influence on the outcome of the games in most cases. The solutions of N-person games with quadratic cooperation functions show drastic changes in some very narrow parameter ranges. The explanation of this phenomenon would require a thorough mathematical analysis of N-person games, which is not yet available in the literature. Further simulations of N-person games may also lead to their better understanding.