Payoffs

Next to the different indicators for stability, the payoffs obtained in the matchings resulting from the simulations are also an important measure of the optimality of a matching. In this section we investigate the payoffs obtained by the agents during the simulations and compare

5.4 Simulations in our model with different parameter settings 53

Fig. 5.11 The average number of blocking pairs over 200 simulations after 100 daterounds for 10 male and 10 female agents

them with the payoffs obtained in the optimal stable matchings. Without loss of generality we only investigated the payoff of the male agents in our simulations because of the duality between the two sides of the market (in our simulations there is no proposing or accepting party, as opposed to the Deferred Acceptance Algorithm) .

In a random matching i.e. after one dateround, for all values for pandswe find average

payoffs of 5. During the simulations we find that the highest average male payoff is obtained

when all agents have independent preferences (p=0) and when it is more likely that an

agent is liked by agents that they like (s=1), with 7.5 on average after 20 daterounds (Figure

5.18), 8.1 after 60 daterounds and 100 daterounds (after 44.7 daterounds on average the optimal stable matching is obtained in these cases). The payoffs of the male agents are lowest

when all male agents fight for the same partners (p=1). Since payoffs are based on random

generated real numbers between 0 and 10 and since all agents of one side of the market have the same preferences over the agents of the other side of the market, the average payoff obtained by the agents cannot exceed 5. When we continue the simulations until a stable

matching is obtained then we find maximal payoffs for p=0 and s=1 around 8.15. For

p=0 ands=0 the payoffs are 7.28 and for p=1 they are trivially around 5, independent

54 Understanding the model

Fig. 5.12 The average percentage of simulations in which a stable matching was obtained within 20 daterounds for 10 male and 10 female agents over 200 simulations

Payoffs of male agents in a relationship

When we only look at payoffs of the male agents who are in a relationship then we find that in the beginning of the simulation, for example at dateround 20, we find average payoffs of

around 5.4 for p=1, independent of the value ofs. For p=0 ands=1 we find that the

average payoffs of all male agents in a relationship are 8.3. We can thus conclude that the pairs in which relatively high payoffs are obtained match earlier in the simulation.

Payoffs of model compared to those obtained in the optimal stable matchings

When we run simulations until the simulations have converged to stable matching we find

that for p=1 and any value forsand fors=1 and any value for pthat we trivially have that

all agents obtained 100% of the payoff obtained in the two stable optimal stable matchings

since there exists only one stable matching in these cases. For p=0 and s=0 we have

multiple stable matchings, and our simulations do not always converge to one of the optimal

stable matchings. Forp=0 ands=0 we find that the male agents obtain 94% of the payoffs

5.4 Simulations in our model with different parameter settings 55

Fig. 5.13 The average percentage of simulations in which a stable matching was obtained within 60 daterounds for 10 male and 10 female agents over 200 simulations

we however find that the male agents obtain 111% of the payoffs they would obtain in the optimal stable matching for female agents (Figure 5.20) and (Figure 5.21).

Higher average payoffs in model than in Deferred Acceptance Algorithm

We compare the average payoff obtained in our model, to the average payoff obtain over the two optimal stable matchings resulting from the Deferred Acceptance Algorithm. We do this by expressing the average obtained payoffs in our model as a percentage of the payoff that would have been obtained by the same agents in the stable matchings resulting from the Deferred Acceptance Algorithm.Very interestingly we now find (Figure 5.22), that the matching to which our model eventually converges, yields higher payoffs than the optimal stable matching for male agents and the optimal stable matching for female agents do on

average. For p=1 and any value for sand for s=1 and any value for pour model, of

course, converges to the same stable matching as the optimal stable matchings that result from the Deferred Acceptance Algorithm with both the male agents and the female agents

as proposing party. However, the lower pand the lowers, the relatively higher the obtained

56 Understanding the model

Fig. 5.14 The average percentage of simulations in which a stable matching was obtained within 100 daterounds for 10 male and 10 female agents over 200 simulations

for p=0 ands=0. This means that our model on average gives us a stable matching with

higher average payoffs than the Deferred Acceptance Algorithm would.

Increasing the number of agents

Increasing the number of agents has no interesting effects on the obtained payoffs. We do find

that for low values for pwe eventually obtain higher payoffs when we increase the number

of agents. When there are more possible partners, agents have “more to choose from", and when preferences are not totally objective, this results in an increase in the obtained payoff. For example 8.15 is maximal payoff for 10 male and 10 female agents, where this is 8.8 for

20 male and 20 female agents (both for p=0 ands=1). On the other hand it of course

takes longer for these relatively higher payoffs to be obtained since it takes longer to obtain a stable matching.