Encounters between AG and PD players in large groups

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Let us turn to another type of situation where the number of players is significantly large. In such a situation, members meet anonymously, they cannot remember the exact course of actions followed in the past by any particular player, yet past aggregate outcomes are observable and remembered. In these conditions, agents have no incentive to build up a 'good' reputation and, therefore, to play strategically has not the same meaning as when group size is restricted. To proceed with the analysis of such games, let us first consider the payoff matrices described in Figure 5.22: the first one gives the benefits accruing to an AG player, when the proportion of players who cooperate varies from 0 to 100 per cent while the second one gives the benefits accruing to a PD player in the same circumstances.

The argument behind this example is the following. In an N-person game, the gains from cooperation and defection for each actor obviously depend on the proportion of people who actually co-operate (or defect) in the entire group. The gains which both AG- and PD-type players derive from co-operation decrease when the proportion of co-operating members in the group declines. Yet such gains are higher for AG players than for PD players for any given proportion of co-operators in the group. On the contrary, the gains from defection are always smaller for AG players than for PD players according to the proportion of co-operators in a large group players. Moreover, the latter's gains from defection have a tendency to diminish with the proportion of co-operators in the group: it is more rewarding to free-ride when everyone else cooperates than when only a fraction of the other members co-operate, and the gains from free riding are at their lowest when defection is generalized.

FIG. 5.22. Payoffs to AG and PD type

By contrast, the gains from defection accruing to AG players exhibit a constant pattern even when the proportion of co-operators in the group decreases. This is because two opposite effects are at work when these players defect. On the one hand, there is the above-noted fact that defection is all the less rewarding as the percentage of tree riders in the population increases. But, on the other hand, AG players 'feel had' about detecting, especially so if they are amidst a large number of co-operating people. Or, to put it in the converse way, the higher the proportion of free-riders in the group, the more the! are relieved of their 'bad feelings' since they can justify their 'opportunistic' acts by reference to the fact that many others behave in the same way as the! do. Consequently the net effect of an increase in the proportion of free-riders on the utility payoffs accruing to AG players when they defect cannot be determined on an a priori basis. Here, we have assumed that the two effects exactly counterbalance each other so that these payoffs are left unaffected by changes in the percentage of freeriders in the group.

Furthermore, it is worthy of note that the payoff to AG players when they cooperate and everybody else also co operates (or when more than 60 per cent of all members co-operate) is higher than the payoff they receive when they are the only ones to defect in the group (6 units): this is a typical reflection of an AG-preference structure. The opposite is of course true of PD players who receive higher payoffs by defecting than by co-operating not only when all other members or a majority of them co-operate but also when few others or even nobody in the group co-operates. Another noteworthy feature is that the payoff received by PD players when they freeride jointly with everybody else (8 units) is smaller than that which they obtain by co-operating jointly with everybody else (10 units), a feature characteristic of a PD game. This, of course, holds a fortiori true for AG players.

It is immediately apparent from Figure 5.22 that PD players have a dominant strategy which is to defect. As for AG players, their preferred strategy will obviously depend on their expectations regarding the likely behaviour of the other players. They will choose to co operate if they expect more than 60 per cent of the group members to co-operate, otherwise they will defect. Thus, for example, if AG players assess the proportion of co-operators in the group to be around one-half, generalized freeriding will take place as both types of players choose to defect. In this kind of situation, the meaningfulness of the concept of trust is evident. In the words of Dasgupta, trust here is to be understood 'in the sense of correct expectations about the actions of other people that have a bearing on one's own choice of action when that action must be chosen before one can monitor the actions of those others' (Dasgupta, 1988: 51).

The main conclusion that emerges from the above N-players game at this stage is the following: for co-operation to prevail on a large scale in an anonymous society or in a large group, it is not sufficient that a significant majority of people prefer universal co-operation but it must also be the case that these people feel confident enough that their willingness to cooperate is shared by many others too.

Now the question is not only how, or under what conditions, collective action can occur in a large group with the characteristics considered here; the question is also whether the cooperative outcome can be sustained on a large enough scale over time. To answer this last question, more information is needed about the dynamics of expectation formation. In a dynamic setting, indeed, decision by AG players whether or not to co-operate requires continual reevaluation of the probability that others will also co-operate based on concrete experiences in past rounds. Not only do expectations affect co-operative behaviour but, over time, past cooperative outcomes affect expectations and future actions, though in a way that leaves no room for strategic considerations: a single player's co-operation cannot affect the proportion of cooperators in the group.

In accordance with what has been said above about the observability of past aggregate outcomes, the assumption is made that agents are broadly able to make out ex post whether and to what extent the collective action under concern has been successful. This is because they can observe the concrete results that collective action has produced: an irrigation canal has been more or less well maintained; foreign trawlers have been effectively deprived of access to inshore waters; the spawning area for fish has not been encroached upon; no felling of trees or cutting of wood has happened in the forest during forbidden times; little grazing occurred on the collective fields before the date fixed, etc. As is evident from these illustrations, the members of a large group may even be in a position to approximately assess the relative number of individuals who have co-operated or defected (yet they are not able to personally identify them).

Let us adopt the following conventions:

P^AG denotes the proportion of AG players in the group;
P^PD = 1 - P^AG denotes the proportion of PD players in the group;
P* denotes the minimum proportion of co-operators required to indue co- operative behaviour among AG players;
P_t^e denotes the proportion of co-operators whom AG players expect to be present in the group at time t + 1; (P_e⁰ is therefore the initial expectation of AG players which reflects their beliefs about the percentage of group members who will co-operate in the first round of the game)
P_t^a denotes the actual proportion of co-operators in the group at time t.

We know that, if P₀^e ³ P*, AG players choose to co-operate at the beginning of the game and, as a result, the actual proportion of co-operators equals the proportion of AG players in the group: P^a₁ = P^AG. On the other hand, if P^e₀ < P*, AG players choose to defect and P^e₀ = 0.

We are now ready for a discussion of the dynamics of collective action in a large group where there are two types of players with the preferences depicted in Figure 5.22. Four possibilities can be distinguished. Under the first possibility, we have P^AG _³ P^e_{0 ³} P*. The AG players co-operate from the beginning of the game, P^e₁ is equal to P^AG for all t greater than zero, and their willingness to so behave is actually confirmed as more rounds are completed. If P^e₀ is strictly smaller than P^AG, these players realize after the first round that the actual proportion of co-operators in the group is higher than what they had initially expected (bear in mind that P^a₁ = P^AG since P^e_{0 ³} P*). Consequently, their expectations are revised upwards and P^e₁ becomes equal to P^AG at t = 1. If P^e₀ is equal to P^AG, AG players discover after the first round that their expectations are fully justified by experience and no change occurs in their expectations. In both cases, collective action is clearly a durable outcome.

The second possibility arises when the following conditions are satisfied: P^e₀ > P^{AG ³} P*. This is typically the case where AG players are overoptimistic about the likely behaviour of others, yet this does not prevent collective action from being established and sustained. The AG players participate in collective action but they are led to bring down their assessment of the likely proportion of co-operators in the light of the first round's experience.

Such is not the case under the third possibility where the overoptimism of AG players cannot avoid the collective action to suddenly collapse at the second round. This case obtains when we find P^e_{0 ³} P* > P^AG The problem obviously arises from the fact that there arc now in the group less AG players than required to induce sustainable co-operation ( P^AG < P*) After the first round, AG players choose to discontinue cooperation forever.

The fourth possibility is the most interesting one. It arises when the proportion of co-operators expected by AG players is smaller than the minimum required to induce co-operation among these players, that is, when P^e₀ < P* < P^AG. In this case, nobody cooperates in the initial round and nobody will ever be incited to co-operate thereafter. In other words, even though there are actually enough willing co-operators in the (large) group to make co-operation possible, such co-operation fails to emerge because they do not have sufficient confidence in the group's inclination to co-operate. Because it cannot be corrected through a co-ordination mechanism, pessimism turns into a self-fulfilling prophecy. This case illustrates the critical importance of trust for cooperation to be possible in large groups.

Note that, even if there is one fully informed AG player who knows that there are actually enough players like him in the population to sustain co-operation, he will not choose to cooperate in the first round since, given the large size of the group, he is unable to persuade others to change their expectations and modify their behaviour. It would be wrong to think that such a result obtains because this individual player is alone to hold correct expectations. To see this, let us assume that, among AG players, there is a subgroup of players who hold optimistic expectations. These players are called subtype I AG players and are distinguished from another category called subtype II who are pessimistic. By optimists, we mean AG players who believe that the proportion of subtype I players in the population is at least equal to P*. Pessimists are those AG players for whom the proportion of subtype I AG players is less than P*.

Two different situations can arise. In a first case, the actual proportion of optimists in the population is higher than P*. After one round, they realize that they are numerous enough to sustain co-operation, no matter what the pessimists do, and the latter are then led to revise their expectations upwards. From the second round Onwards, the pessimists join the optimists in the collective action. The presence of the optimists, to paraphrase Elster, appears as a catalyst for co-operation while the pessimists act as a multiplier on the co-operation of the former (Elster, 1989a: 205). In the second case, the actual proportion of optimists in the population is lower than the critical level P*. After one round when the optimists realize that they are not numerous enough to justify co-operation, and are unable to drive the pessimists in the collective action, they stop co-operating: universal defection ensues.

A richer picture of reality obtains when the assumption of two homogeneous subtypes of AG players is relaxed and replaced by the more realistic one that the degree of optimism of each player is different and unknown to the others. To put it in another way, the distribution of subtypes (i.e. optimism) among AG players is not known a priori. However, the analysis of such a situation lies beyond the scope of the present work. We shall here restrict ourselves to pointing out the main results which can be intuitively expected from such an analysis. The important point to note is that the revision of expectations now takes place in a gradual way after each round rather than in a discrete manner after the first round only. In a border case, all AG players start by co-operating and continue to co-operate forever since even the pessimists have high enough expectations to give co-operation a try. Experience confirms them in their behaviour. A more general case is when the most optimistic players start by cooperating but it turns out in the initial rounds that their number is too small to make co-operation worth while even for them. If those players are led to revise their expectations downwards, some initially pessimistic players may now be induced to cooperate. In such circumstances, it is impossible to say a priori whether co-operation will spread or gradually unravel. Note that in the latter, general case, the most favourable scenario occurs when co-operation is initiated by the most optimistic AG players, then, after subsequent rounds these players revise downwards their expectations yet still cooperate and they are joined by successive batches of players who were initially less optimistic than themselves.

Let us now return to the case where the AG players are divided between two subgroups. However, instead of assuming that members from subtypes I and II differ in terms of the more or less pessimistic character of their expectations, it is possible to differentiate them in terms of the intensity of their interest in co-operation. More precisely, we may assume that players from subtype I derive a higher payoff from cooperation than players from the other subtype, with the result that the threshold proportion for co-operation is lower for the more co-operation-interested players. Let us denote this assumption by writing P*" > P*'. Three interesting cases may be distinguished which lead to results analogous to those obtained in the above analysis of heterogeneous AG players. In a first situation, we have (assuming that players of the two subtypes have similar expectations):

P* < P^e₀ < P*^II < P^AGI,

where P^AGI stands for the proportion of subtype I AG players in the population. Under these conditions, all AG players participate in the collective action after the first round. Players I participate from the very beginning while players II first choose to defect but, as their expectations are being adjusted upwards, concrete experience from the first round gives them enough assurance of others' willingness to co-operate for themselves to join the collective action. This is the virtuous situation in which the more co-operation-inclined players succeed in anonymously persuading the less co-operation-inclined (but non-opportunistic) players to participate in collective action. Thanks to this demonstration effect, the former see their payoffs increase once the latter have joined them. Ex post, we can reinterpret the utility 'losses' incurred by players I during the first round as the necessary price to pay for dragging more prudent men of goodwill into the production of a public good, and thereby draw higher benefits from their own participation in this effort.

A second interesting situation obtains when the following conditions are satisfied:

P*¹ < P^e₀ < P^AGl < P*^ll

Here, the more co-operation-interested players continuously co-operate but, contrary to what we observed in the previous situation, they are not able to prevent the less co-operationinterested players from defecting. This is because, even though the latter's expectations are adjusted upwards, the threshold proportion P*" will not be crossed. Such a situation is especially unfortunate if

P^AG > P*^ll

that is, if the proportion of all AG players in the group actually exceeds that required to induce co-operation among the less co-operation-interested players.

There then remains the third, vicious case where even players I's willingness to cooperate unravels. This case is observed when

P^AGI < P*^l < P^e₀ < P*^ll

which conditions can also be satisfied when P^AG > P*^II Players I start by co-operating but, as players II do not join hands with them, the actual proportion of co-operators ( P^AGI) is too small to incite even the former to sustain their co-operative efforts.