Comportamientos de Riesgo y hábitos de consumo.

In all three models so far it has been assumed that an opportunity for interaction is just presented and any pair of agents can interact. By contrast, in this Section and the next, I will assume that one randomly chosen agent discovers or is endowed with the possibility to interact. In other words, this agent owns the interaction opportunity and will search the set of his potential trading partners to find someone to interact with, in the same way as agent B did in Section 2.1

In this section I assume two groups of agents. As in Section 2.2, all interactions take place between agents from different groups. So if an agent inA (B) discovers an interaction opportunity, he looks round groupB(A) asking each in turn, starting with the Soft agents, until he finds a partner who is available to interact. The assumption made here is that agents in both groups can discover opportunities and all agents have the same probability of being the one to discover an opportunity. If we made

the converse assumption that only agents in one group, say B ever discovers the

opportunities then this is equivalent to the model already analysed in Section 2.1, since, trivially, all agents in group B have the incentive to behave Hard.

Model

There are NA agents in group A and NB agents in group B. Every agent can

adopt one of two behavioural types{S, H}and so the state space is{0,1, . . . , NA} × {0,1, . . . , NB} = X with typical element x = (xA, xB) denoting that xA of group A

and xB of group B are Soft, while the remaining agents are Hard. The state can

change via an evolutionary dynamic which picks a random agent each period who re- vises strategy via the best response dynamic with probability 1−ε, and by mutation with probability ε. All this is the same as in Section 2.2.

Payoffs

Given agents’ behavioural types, the expected payoffs to agents are determined as follows: nature randomly selects an agent who becomes endowed with an opportunity, enabling him to split a surplus of size 1 with a partner. This agent looks for a partner amongst agents of the other group, asking each in turn, until he finds an agent with whom he can interact. As before, he asks all Soft agents before all the Hard ones, with the ordering between any two agents in the group of the same type being random. If the agent with the opportunity finds a partner from the other group to interact with, the two of them split the surplus according to their behaviours as before: equal split if behaviours are the same, or (s,1−s) where s∈(0,1_{/2) in favour of the Hard agent} if their behaviours are different.

Clearly this can be thought of as a zero sum game, since the sum of all agents’

expected payoffs is NB NA+NB 1−(1−p)NA + NA NA+NB 1−(1−p)NB , the probability that an interaction takes place. As in Section 2.2, the agents in your own group are your competitors, whom you wish would act Hard, while the agents in the opposing are your trading partners who you wish would act Soft.

Result: Mixed population possibilities

Agents have two chances to interact. The first comes from possibility of being the

one to discover the opportunity, which happens with probability _N 1

A+NB. The second

comes from the possibility that a member of the opposing group discovers the oppor- tunity and selects you to interact with. The payoffs agents get is the sum of these two possibilities. The effects of the second have been discussed at length in Section 2, where it was found that there is pressure supporting both intra- and inter-group herding. While the effect of the first possibility is simply to create a pressure towards Hard behaviour, since given any behaviour mix in the opposing population, acting Hard adds an extra 1₂ −s on to the average share.

Given this, one might logically expect the herding results of Section 2.2 to persist here. However this is not the case. As the following example demonstrates, the distortion from the first possibility has a large enough effect to move us away from even intra-group herding.

Example 43. LetNA= 4, NB = 2, p= 0.9,s= 0.15.

There are two Nash equilibria and absorbing states here. These arex= (xA, xB) =

(2,2) and (0,0). In other words, it is possible that half of population A are Soft and the other half are Hard. Figure 3.3.2 on page 136 illustrates the evolutionary dynamics over the state space.

Looking at the basins of attraction, we could apply stochastic stability. The tran- sition (0,0) → (2,2) requires two mutations, whereas (0,0) ← (2,2) only requires one. This shows that (2,2) is not stochastically stable, and (0,0) is the LRE. Al- though, in general it is possible for states without herding to be stochastically stable. If s is increased to 0.2, making Soft more attractive, we have a reversal of four ar- rows (in favour of more Soft agents, as one would expect). Under this new dynamic the absorbing states would be {(0,0),(3,1)}. Now, transitioning between the two in either direction would only require one mutation and so both absorbing states are

𝑥𝐵 0 0 1 1 2 2 3 4

Denotes absorbing state

State space and dynamics for

𝑁𝐴 = 4, 𝑁𝐵 = 2, 𝑝 = 0.9, 𝑠 = 0.15

𝑥𝐴

Denotes changes when increase s to 0.2

Figure 3.3.2: Example: NA = 4, NB = 2, p= 0.9.

stochastically stable.

The reasons for this break from herding behaviour may not at first be entirely clear. Indeed much of the intuition from Section 2.2 carries over: The effect of agents in the other group is unambiguous. Being Hard instead of Soft will always increase one’s share when interacting by 1₂ −s. The relevance of the other group’s behaviour is in determining the base from which this increase occurs. When the opposing population is mainly Soft, this increase is from a higher base and so represents a lower propor- tional change compared to when the opposing population is mainly Hard. Thus, just as in Section 2.1, there is a pressure toward inter-group herding of behaviour

The effect of other agents’ behaviour in one’s own population is more complicated. By the analysis of Section 2, taking into account only the expected utility from mem- bers of the other group, we get pressures toward intra-group herding. However, we also have to consider an agent’s expected utility from the chance of being the one to dis- cover the opportunity. This adds an extra _NA+1_NB 1₂ −s 1−(1−p)Nito the util- ity of a Hard type to an agent in group j 6=i, i, j ∈ {A,B}. While this is a constant, it still has an affect: The analysis of Section 2, shows that the ratio of the utilities

Un(S)/Un(H) is increasing inn, the number of one’s own group who are Soft. How-

ever asnbecomes large, bothUn(S) andUn(H) are decreasing and so their difference,

Un(S)−Un(H) can fall. IfUn(S)−Un(H) drops below_NA+1_NB

₁

2 −s 1−(1−p)

Ni

then being Hard is preferable.

Intuitively, what is happening is the following: when enough other agents in one’s group are Soft, the utility one gets from interacting when members of the other group discover opportunities diminishes, whether that agent try to compete by acting soft or not. So it becomes in the agent’s interests to put all his eggs into the basket of maximising payoff from the times when he discovers the opportunity, which means being Hard.

3.3.3

Owning opportunities: one group