Restricted Incentive Compatibility

PART II Toward Efficiency: Coalition Formation Mechanism 65

VII.2 Restricted Incentive Compatibility

It is well-known that when payments are not allowed and/or utility is not transferable, in- centive compatibility is, in general, in conflict with social welfare optimality [141]. One classical approach to address this tension is to additively relax incentive compatibility, re- quiring instead that no player can gain from lying about preferences more than a small amount, ε [142]. This relaxation is justified by suggesting that agents typically face im- plicit costs (actual or cognitive) from lying, or gaming the system.

However, traditional concepts which focus on small gains from lying do not account for another barrier to manipulation: complexity. A common approach in this vein is to consider the computational hardness of manipulation [143]. However, when manipulators arehuman, computational complexity may not be appropriate. We propose several alterna- tive notions of cognitive salience in considering the space of possible manipulations. The

central idea is torestrict the space of feasible manipulationsto those which are cognitively natural. We present three classes of such restrictions:promotion-one IC, where salient ma- nipulations involve promoting a player to the top position; promotion IC, in which agents considerpromotinga prospective roommate in their reported ranking to an arbitrary posi- tion; and the far more general permutation IC, in which agents may report a permutation of their true preferences.

VII.2.1 Promotion-One Incentive Compatibility

We now present the first restriction of manipulating to allow players to promote anyone to the first position in their preference ranking. We term this promotion-one incentive compatibility (POIC).

Definition VII.2.1. A mechanismM ispromotion-one incentive compatible (POIC)for a profile domainD if for any profile∈D, i,j∈N,M()_iM(_i^j,_−i).

Translation to cardinal preferences is direct.

I now present one of our main results, which demonstrates the value of POIC as a re- striction of incentive compatibility. In particular, I show that the most common mechanism for two-sided matching, deferred acceptance [46], is POIC.²

Theorem VII.2.1. Deferred acceptance mechanism is promotion-one incentive compati- ble.

Proof. Without loss of generality, we consider the women-proposing deferred acceptance mechanism (DA for short). We have known that DA is incentive compatible for the propos- ing side (i.e. women), so we will show that it is also promotion-one incentive compatible for the men.

The proof is by contradiction. Suppose that a manmmatches with a womanwunder the true profile, and matches with a more preferred womanw⁰(i.e. w⁰_mw) when reporting

2[37] shows a similar result.

some promotion-one manipulating preference^w_m^∗, where the promoted woman isw^∗. As DA()6=DA(^w_m^∗,_−m), there must be a round of DA that differs when DA is applied to and(^w_m^∗,_−m). Letr^∗be the first such round.

Let S_m denote the set of women held by m, which includesm’s current mate and un- matched women that propose to m in the round r^∗. As r^∗ is the first round in which the outcome of DA is different in the two profiles, we know that the setS_m is the same at the beginning of roundr^∗under both profiles. Becausemis the only player whose preferences are different in the two profiles, m must be the man whose mate is different in the two profiles at the end of roundr^∗. Assume ˆwis the mate ofmunder the true profilein the roundr^∗, based on the property of DA , ˆw_mw, for all ˜˜ w∈S_m.

Also, because w^∗ is the only woman whose ranking changes in _m and ^∗_m, set S_m must contain womanw^∗. (Otherwise, mwould match with the same players at the end of round r^∗ under both profiles, contradicting the fact that DA is different in the round r^∗.) Furthermore, w^∗ must be the woman thatm matches at the end of roundr^∗, andw^∗6=w.ˆ (Otherwise, because S_m is the same under both profiles and w^∗ is the only woman the ranking of which changes in the preference ofm,mwould again match the same players at the end of roundr^∗under both profiles.) Then we could know that ˆw_mw^∗.

Asw^∗has been promoted to the first place inm’s preference^∗_m,mcannot expect to get a better mate thanw^∗. At the same time, the receiving side players can only improve (or re- main the same) with respect to their reported preferences as the rounds of DA progress. So DA_m(^w_m^∗,_−m) =w^∗, andDA_m()_mw. It impliesˆ DA_m()_mwˆ _mw^∗=DA_m(^w_m^∗ ,_−m), which contradicts with the assumption thatmmatches with a better player by re- porting preference^w_m^∗.

This result may help explain the success this mechanism has had in practice, with rather little concern about preference manipulation: it appears that it is incentive compatible under ahighly salientset of preference manipulations.

VII.2.2 Promotion and Permutation Incentive Compatibility

POIC is still a rather restrictive set of manipulations, and it is natural to consider further generalizations. The first generalization is promotion incentive compatibility, where pro- motion can be to an arbitrary position in the preference ranking. To formally define it, let _i^j→l denote a manipulation of an original preference ranking ofi, _i, in which j is pro- moted to a positionl. Let p(_i,j) be the position of j ini’s original preference ranking.

Suppose that position l <k in a preference order _i means that a player in position l is more preferred than one in positionk.

Definition VII.2.2. A mechanismM ispromotion incentive compatible (PIC)for a profile domainD if for any profile∈D, for every player i, prospective partner j, and position l<p(_i,j),M()_iM(_i^j→l,_−i).

The final relaxation of IC we consider ispermutation incentive compatibility.

Definition VII.2.3. A mechanismM ispermutation incentive compatible (Permutation IC) for a profile domainD if for any profile∈D, for every player i,M()_iM(⁰_i,_−i), in which⁰_iis any permutation of_i.

Again, we can translate these two definitions to cardinal preferences directly. Observe that if players are required to submit full preference orders among feasible roommates, permutation IC is equivalent to general IC; consequently, this is a very general notion of incentive compatibility.

Perhaps surprisingly, Example 2 in [114] shows that deferred acceptance mechanism is not evenpromotion IC. Thus, our positive result above is tight.

One of the major successes of the deferred acceptance has been the belief that it doesn’t incentivize manipulation in practice[144]. Showing that it is IC for the restricted subset of promotion-one manipulations, coupled with this anecdotal claim, offers evidence that promotion-one manipulations may be the mostsalientmanipulations in matching settings.

Consequently, our experimental evaluation below focuses on promotion-one incentives to measure benefits from misreporting preferences.