Stability

The most important notion of the stability of a matching seems to be the number of blocking pairs. We also investigate the percentage of simulations in which a stable matching is obtained and whether this is a suboptimal stable matching or one of the optimal ones resulting from the Deferred Acceptance Algorithm. Furthermore, we investigate the number of agents that have the same partner as they would have in the optimal stable matching for male agents and for female agents.

Blocking pairs

When we have obtained random matchings after 1 dateround for 10 male and 10 female agents we find that the average number of blocking pairs, thus pairs of agents who would rather have been matched with one another than with their current partner, varies between 22

48 Understanding the model

Fig. 5.6 The average percentage of agents in a relationship over 200 simulations after 60 daterounds for 10 male and 10 female agents

and 31 (Figure 5.8). The random matchings that are least stable, and thus have the highest

number of blocking pairs, are found for high values ofsand low values of p. This is the case

since when we have high values forsit is more likely that when an agentaprefers being in a

relationship with an agentbthat is not the current partner of agenta, it is also more likely

that agentbwould also want to be in a relationship with agenta. In the same manner, for

lower values of prandom matchings are less stable, since agent’s preferences differ more.

Since when agents have more similar preferences, preferences become less symmetric since if for example every male agent likes the same female agent most, it can of course not be the case that all these male agents are all the most preferred male agent by this female agent.

Interestingly we find that after 20 daterounds that fors=1 andp=0 we find the most stable

matchings with only 3.2 blocking pairs on average (Figure 5.9). The matchings are least

stable for p=1, with around 10 blocking pairs on average, independent of the value ofs.

It seems to be the case that for high values ofsand low values of pwe converge towards a

stable matching fastest. This is the case because agents all have different preferences and it is very likely that when an agent wants to start a new relationship with an agent he or she meets it is also more likely that this is the case the other way around. After 60 (Figure 5.10)

5.4 Simulations in our model with different parameter settings 49

Fig. 5.7 The average percentage of agents in a relationship over 200 simulations after 100 daterounds for 10 male and 10 female agents

and 100 (Figure 5.11) daterounds we find that matchings are least stable for p=0 ands=0

with on average 4.1 and 2.8 blocking pairs respectively.

Percentage of simulations in which a stable matching is obtained

We have investigated how many of the simulations for 10 male and 10 female agents have reached a stable matching after 20 (Figure 5.12), 60 (Figure 5.13) and 100 (Figure 5.14) daterounds and investigated how many of those stable matchings were suboptimal and how many were optimal for either the male or the female agents or both. We found that most

stable matchings were obtained for high values forsand low values for pwith 7% after 20

daterounds, 34% after 60 daterounds and 99% after 100 daterounds. For p=0 ands=0 the

least stable matchings are obtained.

When we run simulations until our model has converged to a stable matching Figure 5.15

we find that a stable matching is obtained fastest forp=0 ands=1, with 44.7 daterounds

on average. When there is no symmetry and all agents have different preferences (p=0 and

s=0) it takes the most rounds to obtain a stable matching with 161 rounds on average. For

50 Understanding the model

Fig. 5.8 The average number of blocking pairs over 200 simulations in a random matching after 1 dateround for 10 male and 10 female agents

Percentage of simulations in which a suboptimal stable matching is obtained

Not many of the obtained stable matchings for 10 male and 10 female agents are suboptimal, meaning a stable matching other than one of the optimal stable matchings that would result

from the Deferred Acceptance Algorithm. We find that for low values for pandsaround

1% of the simulations ended up in a suboptimal stable matching after 20 daterounds. After

60 daterounds this percentage has increased, with a maximum of 8% for p=0 ands=0.

After 100 daterounds this percentage has increased to 13%. Eventually, after 161 daterounds

(Figure 5.16) on average we find that 35% of the 200 simulations fors=0 andp=0 ended

in a stable matching that is neither optimal for the male agents nor for the female agents. For

p=1, independent of the value forswe never obtain a suboptimal stable matching. This is

also the case fors=1 and any value of p.

We have proven Theorem 9 that said that in any Stable Marriage Problem in which all

agents of one side of the market have the same preferences over the other side of the market

and the other way around (p=1), there exists only one stable matching. From this proof

5.4 Simulations in our model with different parameter settings 51

Fig. 5.9 The average number of blocking pairs over 200 simulations after 20 daterounds for 10 male and 10 female agents

Problem in which all agents of one side of the market have the same preferences over the other side of the market and the other way around.

We have proven Theorem 10 that said that in any Stable Marriage Problem with symmetric

preferences (s=1), there exists only one stable matching. From this proof trivially follows

that for any instance of Stable Marriage Problem with symmetric preferences no stable matching exists that is not optimal for one side of the market.

Percentage of agents in optimal stable matchings

We also kept track of the percentage of the agents that were matched with the same agents as they would be with in the optimal stable matching for male agents and the optimal stable matching for female agents during the simulation. The average percentage of male agents matched with the same partner as they would be with in one of the optimal stable matchings

is lowest for p=0 and s=0 with both 44% of the male agents being matched with the

same partner as they would be in the optimal stable matching for male agents and 44% of the male agents being matched with the same partner as they would be in the optimal stable

52 Understanding the model

Fig. 5.10 The average number of blocking pairs over 200 simulations after 60 daterounds for 10 male and 10 female agents

matching for female agents on average after 20 daterounds over 200 simulations. After 60 daterounds these percentages both increased to 63% and after 100 daterounds to 68% respectively. Eventually about 75% of the male agents are being matched with the same partner as they would be in the optimal stable matching for male agents (Figure 5.17) and of course also about 80% of the agents being matched with the same partner as they would be in the optimal stable matching for female agents on average over 200 simulations.

Increasing the number of agents

Although it takes many more daterounds to obtain stable matchings we find similar effects

of changing the values for pandson the stability in simulations for 20 male agents and 20

female agents.