Co-operation and collectives

Matt Corks & Suresh Naidu

8 December 1999

Abstract

We will perform a game-theoretic analysis of a typical co-operative. We will show that the principles of the co-op movement create the necessary conditions for achieving a stable, equilibrium outcome in an iterated N-player prisoner’s dilemma game.

Introduction

By definition, the purpose of a for-profit corporation is to increase its shareholder’s wealth. A central authority figure (Leviathan) is typically used to maintain order and ensure that the transfer of wealth from consumers to owners happens as efficiently as possible. However, such managers are also in a position to make the corporation run in ways which are personally convenient for them.

An alternative to this is to form a corporation which is owned by its consumers, or a co-operative. In this system, the owners have no incentive to exploit the consumers. If the consumers also manage the corporation, then there will be no opportunity for an external manager to cheat both the owner-consumers for their own gain. Elementary results of N-player game theory show that co-operatives can be successfully run by their members, who are able to work together for their mutual reward.

Co-operatives, and in particular those without paid staff who act as managers, provide a wealth of situations that illuminate principles of N-player game theory. They are a democratic, sustainable alternative to the corporation, which organizes power in a purely hierarchical fashion. The co-operative is an economic entity more amenable to an equitable allocation of resources and control. This is done very simply by creating a ‘tragedy of the commons’ situation between members.

Consider the case of a buyer’s co-op retail store. Work put into the co-op by any member, such as time invested in repairing the store, will increase the value of the store for all members. However, it is in the interests of each member individually to remove more value from the corporation than they have contributed. They will be better off taking this approach if all the other members either contribute to the store or do the same. However, if too many members takes this dominant strategy, the co-operative will be of less value to every member.

This is a classic ‘tragedy of the commons’, and is deliberately created by the principles of the modern co-operative movement, as adopted by the International Co-operative Alliance in 1966. These principles were based largely on the charter of the original Rochdale co-op, and are as follows:

Open and voluntary membership
Democratic control (one member, one vote)
Limited interest on shares
Return of surplus (profit) to members in proportion to the amount of business each member does with the co-op
Co-operative education
Co-operation among co-operatives

A game theory model of a co-op

It is possible to model the members of a co-op as players in an iterated N-player prisoner’s dilemma game. This shows that co-operative behaviour can emerge amongst a group of egoists, and describes the circumstances under which it is most likely.

For each player, define f(v) as the payoff if that player co-operates and v other players co-operate and g(v) as the payoff if that player defects and v other players co-operate.

To make this a prisoner’s dilemma situation we require that:

(1): g(v) > f(v) ∀ v > 0, since we want defections to payoff more than co-operation for each individual
(2): f(N -1) > g(0), since we want the payoff for mutual co-operation to be greater then the payoff from mutual defection
(3): g(v) > g(0), so that the payoff for defecting is always greater than when there are no co-operators

We may assume that (1) is true, since it is always in an individual’s self-interest to ‘mooch’ off the other members of the co-op. (2) follows from the fact that a co-operative business cannot run without some upkeep and investment of time, and (3) since it is impossible to profit from the co-op if no-one is maintaining it.

We will denote strategy vectors as vectors of the form (A₁,A₂,A₃,...,A_N), where A_i is the strategy of the i^th player. The case where the players are all unconditional co-operators is indicated by (C,C,C,...,C). This is clearly not in equilibrium, since any co-operator has a unilateral incentive to defect. The case where every player always defects is given by (D,D,D,...,D). This is in equilibrium, since changing to co-operation here would only make one a sucker.

Our objective is to find strategy vectors that both sustain co-operation and are in equilibrium. We start with a generalization of tit-for-tat we will call TFT_n, which indicates that the player will co-operate iff n other players co-operated in the previous round. We can show that (TFT_n,TFT_n,...,TFT_n) is in equilibrium iff n=N-1. If a player i unilaterally changes to D, then in the first game they get the payoff g(N-1), but they will get g(0) thereafter, and defection will be sustained forever, because there is never enough co-operation to get the TFT_n players to return to co-operation. If n < N-1 then some players can defect while the rest sustain co-operation, so those defectors would increase their payoff unilaterally.

The payoff from a player i co-operating is f(N-1)=((a_i)÷(1-a_i)), and the payoff if they defect is g(N-1)=a_i+g(0)(((a_i)²)÷((1-a_i))), where a_i is the shadow of the future for the player i. So, this is in equilibrium iff a_i ≥ g(N-1) - ((f(N-1))÷(g(N-1a)) - g(0).

Consider strategy vectors of the form

(C,C,...,C, TFT_n,TFT_n,...,TFT_n, D,D,...,D)

Let n_c be the number of co-operators, n_d be the number of defectors, and n_b be the number of conditional co-operators. So, n_c+n_d+n_b = N. Let m = n_c + n_b. If we assume that n is the same for every TFT_n, we can show that equilibrium happens iff n=m-1 for every TFT_n player, as follows: if n=m-1 then the co-operators will always choose to co-operate, and so the game reaches an equilibrium. However, there are two cases where n ≠ m-1: if n < m-1, there is too much co-operation, so a player can defect and the rest of the co-operators will continue to co-operate. This game is not in equilibrium. If n > m-1, there is not enough co-operation, so the TFT_n’s will immediately start defecting after the first round, and it pays for the co-operators to also defect and not become suckers.

A TFT_n gains from a defection iff

f(m-1) ((a_i)÷(1-a_i)) ≥ g(m-1)a_i +g(n_c) (((a_i)²)÷(1-(a_i)²))

The shadow of the future for each player is

a_i ≥ g(m-1) - ((f(m-1))÷(g(m-1))) - g(n_c)

Now, the tragedy of the commons can be represented as problem of collective action. Each user of the commons can choose to co-operate, by not using the goods excessively, or defect by voraciously using the commons. Assume then that the net goods provided is r such that 1 < r < N, and the benefit to each individual is ((r)÷(N)).

Let

f(V) = (((v+1)r)÷(N)) -1

and

g(v) = ((vr)÷(N))

Note that g(v) > f(v) ∀ v ≥ 0 since r < N. As well, f(N -1) > g(0) and g(v) > g(0), since r > 1, so these meet our criteria for an N-player prisoner’s dilemma. If we assume that everyone is playing TFT_n, then by above:

a_i ≥ ((g(N-1) - f(N-1))÷(g(N-1) -g(0))) ∀ i

so substituting for f and g we get

a_i ≥ ((N-r)÷((N-1)r))

Note then that a_i is only dependent on N and r. If a_i → 0 then this is satisfied for r > N as well. But if a_i → 1 (i.e., the long term benefits pay off more), then this is satisfied for any r > 1. This is expected, as if the public benefit for each unit contributed is large, then TFT_n will contribute even if the shadow of the future is very small, and if the shadow of the future is large (i.e. a_i close to 1), then it will pay to contribute even if the benefit is small.

Now consider strategy vectors of the form

(C,C,...,C, TFT_n,...,TFT_n, D,D,..D)

From above we have:

a_i ≥ g(m-1) - ((f(m-1))÷(g(m-1))) - g(n_c)

Substituting again for g and f:

a_i ≥ ((((N)÷(r))-1)÷(n_b -1))

So, this is just dependent on the number of conditional co-operators, N, and the payoff, r, as we would expect.

This analysis, which is due to Taylor, has the flaw that it does not consider a wide range of strategies and so does not show that co-operative strategies can emerge amongst defectors and be evolutionarily stable, in the manner than Sigmund & Nowak have done for the two-player IPD. However, several other researchers, such as Yao & Darwen, have concluded that co-operation can still emerge and be evolutionarily stable in the N-player IPD. Like Taylor, they conclude that it is easier for co-operation to emerge within small groups. In addition, a great deal of research has been done on optimal coalition sizes in the N-player IPD. This is beyond the scope of our report, but it is fair to say that this research has reached the same general conclusion.

Results and Discussion

We have seen that it is possible for co-operation to emerge in a co-operative composed of self-seeking egoists, even if some are unconditional defectors. Further, we have seen that this co-operation requires the presence of members who co-operate only conditionally, since this stops players from making a unilateral move to defection. This in turn implies that co-operation is more likely to emerge in a small co-operative, since the co-operation of each player depends on the actions of less people. It also implies that efficient methods of communication are necessary, so that players know when others are co-operating. We have also seen that it is possible for co-operation to emerge with a small number of unconditional defectors. And, of course, it is essential for the prisoner’s dilemma situation to be iterated, so that players do not simply take their dominate strategy defect.

This would imply that the most successful co-ops are small, communicate well, have members who tend to stay in the co-op for a long period of time. A small number of unconditional defectors may be tolerated. In the model, the conditional co-operators must choose either to co-operate with all or none of the other members, but in actual co-ops, members usually create mechanisms by which such defectors will either profit less from the co-op than other members, or be forced to leave the co-op altogether. This in effect is conditional co-operation.

These principles emerge in real co-ops. For example, the Waterloo Co-operative Residence Incorporated (WCRI) is a housing co-op for students at UW and WLU, which means that people are rarely members for more than five or six years. It is run according to the Rochdale principles, and currently houses 900 students, but its buildings have been described as “a dive” by many tenants and members lack a co-operative spirit. Although each is required to put in from one to three hours a week of work for one of the various committees in the co-op, they generally see these tasks as a part of their rent payment and not as something they can do for the community as a whole. In no other co-operative we know of in town do members resent the work they are asked to do for the co-op as strongly. This situation is markedly different from that at the Beaver Creek Housing Co-op, which is not comprised of students. It is also run as according to the Rochdale principles, but its members tend to speak of the co-op as a community and usually feel that they are contributing to the group when they invest their time in the co-op. They have in their history evicted tenants who did not meet their obligations, and the members speak of this as having strengthened their co-op.

It would be interesting to conduct proper psychological studies of the differences in attitudes between the members of these co-ops. However, our informal interviews of aquaintances led to clear conclusions.

Conclusion

Co-ops are an excellent demonstration of how it is possible to overcome a ‘tragedy of the commons’ without a Hobbesian Leviathan, and all the more so because they have created the dilemma for themselves in full knowledge of what they were doing. The founders of the first co-ops must have arrived at conclusions similar to Taylor, Rapoport, and other game theorists when deciding that co-operation could emerge in a group of egoists. A century and a half later, we see their principles being successfully throughout the world, and not by any accident. The tools of modern game theory show that this movement will continue to succeed and grow.

References

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