N-person, multi-period, meta-preference Prisoners’ Dilemma in an evolutionary game

In the real world, most of the interesting public goods provision dilemmas, including irrigation interactions, involve more than two actors. N-person games, therefore, could produce more practically relevant insights.

We thus model water users’ interaction by considering a water using farmers’ population composed of N individuals who interact in pairs to engage in irrigation investment activities. We simultaneously introduce repetition (of the same game), retaliation (tit-for-tat preference) and replication (of the norms of the successful players) to the N-person Prisoners’ Dilemma and hence show how it leads the interaction to multiple equilibria of both mutual defection (abstaining) and mutual cooperation.

Our modelling strategy follows evolutionary game theory, which is a modified version of the classical game theory that takes into account people’s limited cognitive capacities. Hence individuals, according to evolutionary game theory, update own beliefs, and accordingly decisions, using imperfectly observed local information. Evolutionary game theory can describe adaptive water users who might not necessarily be forward-looking, and whose interactions’ direction is determined by differential replication which then determines the population structure with preferences including both self-regard and reciprocity (Bowles, 2004). More details about evolutionary game theory can be found in Weibull (1995), Bowles (2004) and Dixit & Skeath (2004).

In an evolutionary game setting, the adaptive agents keep updating their choices of traits, and they do it in accordance with differential replications. We assume that at each period of the interactions some 𝜔 fraction

65 of water users’ population update their choice of strategy. The updating dynamic favors successful strategies over less successful strategies. The fitter strategies with higher expected payoffs, as a result, get more replicas (Weibull, 1995). These expectations are simply the payoffs that would obtain if the previous period’s state remained unchanged (Bowles, 2004: 408). This replication dynamic gives direction to the evolutionary processes (Bowles, 2004: 62)

We illustrate this extension of the game, with repetition, retaliation (tit-for-tat) and replication, in Table 3-2, which is adopted from Bowles (2004).

Table 3-2: Payoff table of iterative, multi farmer irrigation investment interaction with retaliation preference possibility

Tit for tat Abstain

Tit for tat ^𝑏_𝜌; ^𝑏

𝜌 𝑑 + (1 − 𝜌)^𝑐

𝜌; 𝑎 + (1 − 𝜌)^𝑐

𝜌;

Abstain 𝑎 + (1 − 𝜌)^𝑐

𝜌; 𝑑 + (1 − 𝜌)^𝑐

𝜌

𝑐 𝜌; ^𝑐

𝜌

Note: 𝑎 > 𝑏 > 𝑐 > 𝑑; 𝜌 ∈ [0; 1]; Source: adopted from Bowles, 2004: 242

We assume, for simplicity, that the N-person population of farmers is endowed with two preferences only.

One is tit-for-tat (T), i.e. the player with such a trait will cooperate in the initial period and in all subsequent periods will do what the counterpart did in the preceding period of interaction. The second preference is unconditional abstaining (A) from investment. We suppose that the players are randomly paired to play after each period of play. Extension of the model also captures the iterative nature of interactions, and it is reflected in a newly introduced element to Table 3-2, that is the probability of interaction to terminate (𝜌).

The range of 𝜌 varies between 0 and 1. The closer it is to 1, the higher is the probability of termination, and the interaction illustrated in Table 3-2 will tend to resemble the one illustrated in Table 3-1. On the other hand, if 𝜌 is closer to 0, the higher is the probability of the game to be repeated and the resulting game resembles an Assurance game. We assume that the repetitions take place over appropriately brief periods and hence justify our ignorance of the players’ discount rates.

We normalize the size of the farmers’ population to unity and denote the fraction of farmers who are retaliating (play tit-for-tat strategy) type with 𝜏. Consequently, (1 − 𝜏) is the fraction of farmers’ population

66 who are (unconditionally) abstaining. The expected payoffs for tit-for-tat and unconditional abstaining players are denoted with 𝜋^𝑇and 𝜋^𝐴, respectively, and they take the following values:

𝜋^𝑇 = 𝜏^𝑏

𝜌+ (1 − 𝜏) {𝑑 +^{(1−𝜌)𝑐}

𝜌 } (1)

𝜋^𝐴= 𝜏 {𝑎 +^{(1−𝜌)𝑐}_𝜌 } + (1 − 𝜏)_𝜌^𝑐 (2)

By equating (1) and (2) we get 𝜏^∗, i.e. the interior equilibrium share of tit-for-tat playing farmers:

𝜏^∗= ^𝑐−𝑑

2𝑐−𝑎−𝑑+(𝑏−𝑐)/𝜌 (3)

Figure 3-1illustrates (1), (2) and (3). In this model, we represent water using individuals as bearers of their adopted strategies (tit-for-tat or abstain). However, the distribution of chosen strategies varies within the population. While analyzing the change in a single period (∆𝜏), we follow the assumption of monotonic updating of the individual strategies. This, in turn, implies that ∆𝜏 takes the signs of (𝜋^𝑇− 𝜋^𝐴) (as in Bowles, 2004:409).

Figure 3-1 is non-ergodic or path dependent, as there are two stable equilibria both of which are absorbing.

Which equilibrium is attained by the population depends on the initial state (Young, 1998:48). This situation 1

0 𝜏^∗

𝑑 + (1 − 𝜌)𝑐/𝜌 𝑐/𝜌

𝑎 + (1 − 𝜌)𝑐/𝜌 𝑏

𝜌

Payoffs

𝜏: Fraction playing tit-for-tat

Figure 3-1: Expected payoff to strategies.

𝝅^𝑨line: expected payoff to abstainers. 𝝅_𝑪^𝑻 line: expected payoff to tit-for-taters

67 is subject to positive feedbacks as the payoff to either strategy (to invest or abstain) is increasing in the number of people taking the same action. Moreover, there is a threshold (𝜏^∗) amount of tit-for-taters, i.e. an unstable equilibrium, beyond which tit-for-tat becomes more successful than abstaining. This is because when the fraction of farmers playing tit-for-tat is more than that threshold, the payoffs to tit-for-tat become greater than the payoffs to unconditional abstaining (𝜋^𝑇− 𝜋^𝐴> 0). In this setting with positive feedbacks, small chance events usually have continuous consequences. Initial conditions produce persistent ‘lock-in’

effects and lead the population into multiple equilibria or ‘traps’ as in Figure 3-1. In such traps (absorbing stationary states at 𝜏 = 0 and 𝜏 = 1), small deviations in strategies (∆𝜏) are not sufficient to shift the interaction from one state to another, unless ∆𝜏 > 𝜏^∗ or ∆𝜏 >1 − 𝜏^∗ respectively. The steady states (equilibria) are self-correcting. However, the multiple stable equilibria can still be displaced by means of exogenous shocks, mutations and non-best response play (Bowles, 2004:12).

3.2.4 Arrangements (treatments) facilitating cooperation

Achieving a cooperative outcome in smaller groups is more realistic than in bigger groups. Taylor (1987:105) justifies the size effect with the argument of peer monitoring, as it is a major enabling factor for players to sustain conditionally cooperative interaction. With increasing group size, however, it becomes a tedious task for the interactors to engage themselves in mutual monitoring, as a result, sole peer-monitoring might lose its worth as cooperation inducing arrangement. Consequently, in groups of intermediate size, positive and negative sanctioning mechanisms could be essential to facilitate the self-reinforcing cooperative outcome.

Until now we have seen how a multi-period, N-person Prisoners’ Dilemma with retaliation turned into an Assurance game-like interaction with multiple Pareto ranked (superior and inferior) equilibria. The Assurance game which is also known as a Trust Dilemma (Grimm et al., 1998:163) can be locked into a defective convention due to a lack of trust among players, as the alternative name might suggest.

68 Pre-play communication may provide the players with trust and hence reputation building opportunity.

Communication among players enables them to behave conditionally cooperative which then (in the following round of interaction) increases the proportion of tit-for-taters in the population. This effect of communication, through its peer-monitoring specification, is reflected in Figure 3-2 by an upward shift in the expected payoff for tit-for-tat denoted with 𝜋_𝐶^𝑇. Such shift decreases the threshold amount of the population fraction of tit-for-taters (𝜏_𝐶^∗ < 𝜏^∗). It implies that the basin of attraction for the cooperative (mutual investment) convention is increased.

Sanctions, on the other hand, diminish the payoffs for unconditional defectors. Figure 3-3 conveys this notion by shifting down the expected payoff of unconditional abstaining (𝜋_𝑆^𝐴< 𝜋^𝐴). This shift, in turn, increases the basin of attraction of the cooperative convention:(1 − 𝜏_𝑆^∗) > (1 − 𝜏^∗). This implies that sanctioning also facilitates the cooperative convention.

𝜏_𝐶^∗ 1

0 𝜏^∗

𝑑 + (1 − 𝜌)𝑐/𝜌 𝑐/𝜌

𝑎 + (1 − 𝜌)𝑐/𝜌 𝑏

𝜌

Payoffs

𝜏: Fraction playing tit-for-tat

Figure 3-2 Expected payoff to strategies. Communication treatment (dashed 𝝅_𝑪^𝑻 line: expected payoff to tit-for-taters in Communication treatment)

69 Figure 3-3: Expected payoff to strategies. Sanctioning treatment (dashed 𝝅_𝑺^𝑨line: expected payoff to unconditional abstainers from investment in sanctioning treatment)

There are hence five major insights to take from our theoretical discussion so far. First, that the repetition of the one-shot Prisoners’ Dilemma makes cooperation possible as a best-response play of rational individuals. Second, the existence of conditional cooperators playing a tit-for-tat strategy is another factor enabling cooperation. Third, the repeated Prisoners’ Dilemma can end up in multiple equilibria. Fourth, the initial state of interaction plays a key role in determining the final equilibrium (i.e., history matters). Finally, peer monitoring (Figure 3-2) and sanctioning (Figure 3-3) may enable conditional cooperation as they make the cooperative convention more attractive (by increasing its basin of attraction).

𝑐/𝜌

𝜋_𝑆^𝐴

𝜏_𝑆^∗ 1

0 𝜏^∗

𝑑 + (1 − 𝜌)𝑐/𝜌

𝑎 + (1 − 𝜌)𝑐/𝜌 𝑏

𝜌

Payoffs

𝜏: Fraction playing tit-for-tat

70