Let four men have combinations of two characteristics: wealth and humor.
• Man 1: {Rich, Funny} • Man 2: {Poor, Funny} • Man 3: {Poor, Boring} • Man 4: {Rich, Boring}
A woman is asked three questions:
• Do you want a partner who is rich or poor? • Do you want a partner who is funny or boring? • Which is more important, wealth or humor?
If poor and funny are chosen as the desired traits, an agent who is rich and boring can never be ranked above an agent who has one of the desired traits
Preference list Acceptable Answers needed
2{P,F} 1{R,F} 3{P,B} 4{R,B} yes Poor, Funny, Wealth is more important than humor
2{P,F} 3{P,B} 1{R,F} 4{R,B} yes Poor, Funny, Humor is more important than wealth
2{P,F} 4{R,B} 3{P,B} 1{R,F} no Impossible
This example shows how the responsiveness in preferences created by the questions limits the preferences that can be represented once a particular set of characteristics and questions is chosen. A dierent three questions over dierent characteristics could allow for the last line of the table to be a possible generated preference list.
Example 3.
The mechanism in this example will be the one that simply maximizes total compatibility scores in the market. A couple score is calculated by taking the minimum of the two individual compatibility scores. The mechanism has many undesirable properties; there is no regard for stability and it is easy to show that the mechanism is not strategy-proof. Additionally, I will simplify the message space to twenty yes or no questions. An agent will answer these questions about himself and what he desires a partner to answer as in the OkCupid matching questions. Each time the partner's answers match, she gains 5% in compatibility score, out of 100% total. There will be three men and three women. Suppose preferences are
m1: w1− 100%; w2− 95%; w3− 60% w1: m1− 95%; m2− 90%; m3− 85%
m2 : w1− 95%; w3− 80%; w2− 75% w2: m2− 100%; m3− 85%; m1− 75%
m3: w2− 100%; w3− 95%; w1− 95% w3: m2− 90%; m3− 85%; m1− 80%
Then the couple compatibility scores are m1w1: 95 m2w1: 90 m3w1: 85
m1w2: 75 m2w2: 75 m3w2: 85
m1w3: 60 m2w3: 80 m3w3: 85
If all agents report honestly, the mechanism will select µ = [(m1w1), (m2w3), (m3w2)], for a maxi-
mized total couple compatibility of 95+80+85=260. Let us consider another matching, µ0, where µ0 =
[(m1w1), (m2w2), (m3w3)]. Here the total is lower, 95+75+85=255, but it is also clear that w2 prefers µ0,
receiving her 100% individually compatible partner, m2, as opposed to m1 at 85% in µ. At this point, we
can observe that w2would like to manipulate her preferences, if possible, to improve her own personal payo
by changing the outcome to µ0.
However, consider one last possible matching, µ”, where µ” = [(m1w2), (m2w1), (m3w3)]. This match
has a total score of 75+90+85=250 and will certainly never be selected by the mechanism if agents report honestly. We can also note that w2prefers µ” the least, receiving her lowest compatible partner, m1at 75%.
It is the existence of this matching that blocks w2's ability to manipulate the market.
Consider w2's strategies if she wishes to manipulate the outcome of the mechanism by lying about her
desired traits. She knows she must lower the total compatibility of µ by at least 10 points for the mechanism to select µ0 instead. Therefore, she must lie twice in her desired traits to drop her reported compatibility
with m3 to 75% instead of 85%. However, because agents cannot directly manipulate their preference lists,
lying about two questions must lower her reported compatibility with m2 from 100% to 90%. Furthermore,
these two lies might increase or decrease her reported score with m1. For this example, supposed the two
lies caused the reported score to increase from 75% to 85%.
As the couple compatibility scores are calculated with the minimum of the two, this changes the couple compatibilities of the pair (m1, w2) to 85% and (m3, w2) to 75%. All others remain the same, including
(m2, w2)as that score is determined by m2's low personal compatibility with w2. With these new couple
new outcome of the mechanism. For match µ, the new total is 250. For match µ0, the new total is 255.
Thus, w2has succeeded in blocking the match µ, as she had hoped.
However, due to the nature of the alternative message space, she cannot simple adjust her reported compatibility with one other man. Because her two lies have changed the reports about all three men, we must also calculate the score of µ”, which is now 260 under her false report. The mechanism will select µ”, resulting in w2receiving her least desirable partner under her true preferences. In this market, the restriction
on the alternative message space prohibits this agent from improving her outcome through manipulation. A similar argument can be made for m2, who also is prevented from improving by deviating in the same
manner. As all other agents are receiving their top choices, they, too, have no incentive to deviate. As no agent can improve by deviating, honest reporting is a Nash equilibrium in this market.
It is worth noting that this Nash equilibrium cannot be sustained under direct reporting. In that case, w2would simply change her reported compatibility with m3, leaving the others unchanged. this would allow
her to block the rst matching and have the second selected. Here, she clearly has incentive to deviate, receiving her top choice by doing so, instead of her second choice.
Proof of Theorem 1.5.
Proof. I rst show that the set of generated preference lists is strictly contained in the set of direct reporting preferences, L ⊂ P .
This follows from the de nition of the two sets.
First, L ⊆ P . L contains at most all permutations of a ranking containing all agents to be ranked. P contains all permutations of any subset of agents to be ranked, including the case when all agents are ranked. Second, L + P . By construction, any item in L must rank all agents. Take any element Pi of P such
that the length of that preference list, or the number of agents it ranks, is less than n. Such an element Pi∈ L/ .
A manipulation in the case of the alternative message space is a report leading to an L0
i(r 0 i) 6= Li(ri). Since any L0 i ∈ P, choose P 0 i = L 0 i.
Thus any manipulation in the alternative message space can be replicated in a direct reporting environ- ment.
A manipulation in the case of direct reporting is any report P0
i 6= Pi. Choose a P
0
i such that it is a
truncation of Pi, and as a result P
0
i ∈ L/ .
Thus the reverse is not true.