PART II Toward Efficiency: Coalition Formation Mechanism 65
VII.3 Automated Mechanism Design for Roommates Problem
Consequently, our experimental evaluation below focuses on promotion-one incentives to measure benefits from misreporting preferences.
full mechanism then becomes Program (VII.2).
PMSW =max
x
∑
i∈N
∑
j∈N
xi jui j (VII.2a)
s.t.:
∑
j∈N
xi j≤1, ∀i∈N (VII.2b)
i∈N
∑
xi j ≤1, ∀j∈N (VII.2c)
xi j=xji, ∀i,j∈N (VII.2d)
IR:
∑
j∈N
xi jui j ≥uii, ∀i∈N (VII.2e) xi j∈ {0,1}, ∀i,j∈N (VII.2f)
VII.3.2 AMD with Approximate Permutation Incentive Compatibility
Next, I consider the problem of maximizing social welfare with restricted forms of in- centive compatibility captured by a collection of constraints. As I had remarked earlier, the central challenge in doing so is that, in general, incentive compatibility must consider mechanism outcomesfor alternative manipulation, something that can be difficult to capture without explicitly defining a general roommates mechanism (a clearly intractable proposi- tion). In this subsection, I demonstrate that the special structure of the roommates problem allows us to overcome this challenge.
As our principle approach, I present an integer linear program that trades off social welfare and an upper bound ofε. And the resulting mechanism isε-permutation incentive compatible. The key idea for developing this approach is to consider each possible manip- ulation by a player i. The worst case, from the mechanism designer’s standpoint, is that the manipulation succeeds, andimatches with a better roommate than her current mate. I can very conservatively guard against all such possible deviations by simply introducing a collection of constraints that each player i obtains at least ui j for any possible partner j they may have. This yields our first integer program for automated mechanism design,
presented in Program (VII.3). In the program,α is set to trade off between social welfare and incentive compatibility and 0≤α ≤1.
PIC=max
x,ε (1−α)
∑
i,j
xi jui j−α|N|ε, (VII.3a) s.t.: constraints (VII.2b)−(VII.2e)
ε-IC :
∑
k∈N
xikuik≥ui j−ε,∀i∈N,j∈Ri (VII.3b)
ε≥0 (VII.3c)
xi j∈ {0,1}, ∀i,j∈N (VII.3d)
The above optimization problem is clearly conservative, as it introduces constraints about prospective roommateswhether or not they can be realized through a unilateral deviation.
I deal with this presently, but for the moment, this provides our first principled approach.
Let us denote byMIC()the mechanism (implicitly) implemented by the integer pro- gram (VII.3). As we now observe, the optimal solution to this program yields an upper bound on the most any player can gain from arbitrary permutations of their preferences—
that is, the result is approximately permutation IC (and, consequently, promotion and promotion-one IC).
Theorem VII.3.1. Assume the optimal solution of Program VII.3 is(x∗,ε∗), then the mech- anismMIC isε∗-Permutation IC, i.e. no player can gain more thanε∗ via permutation of her true preference.
Proof. Due to the IR constraints (VII.2e), playeri never team up with a player that is not inRi. When playeriuntruthfully report her preference by permutation, she will still match with one player inRi(or be a singleton). The constraint (VII.3b) can make sure that player icannot gain more thanε∗by matching with any player inRi.
trade off social welfare and incentives to lie by tuning the parameterα. As our experiments demonstrate, this yields a non-trivial tradeoff with respect to the highly salient promotion- one-IC deviations.
We note that many of the constraints in the above integer program are unnecessary, and, indeed, significantly over-constrain the problem. In the program, all players in the feasible setRi are treated as “potential” teammates wheniis trying to manipulate the mechanism, and some players inRiare far less preferred than the final teammates. Consequently, these constraints will never be relevant.
I use this intuition to develop an iterative approach to generating the approximate per- mutation IC constraints, allowing us to focus only on those which actually matter. We start with a program that maximizes social welfare, and use it to obtain an initial roommates as- signmentπ. Then we add constraints to make sure that playericannot gain more thanεby matching with any player js.t. jiπ(i). We then solve the program with the newly added constraints (replacing the objective with (VII.3a) and adding the constraint thatε ≥0), ob- tain a new assignment, and repeat the process until convergence (which is guaranteed in the quadratic time since the set of possible constraints is quadratic). This approach is shown in Algorithm 10, and the resulting program is, again, ε-permutation IC (details omitted due to space constraints).
Algorithm 10ε-Permutation IC Program input:initial program (VII.2)
return: ε-Permutation IC Program
1: programPIC ←program (VII.2)
2: repeat
3: solve the program and get the matching assignmentπ
4: fori, and j∈Rido
5: if jiπ(i)and corresponding constraint has not been addedthen
6: add∑k∈Nxikuik≥ui j−ε intoPIC
7: end if
8: end for
9: untilthe valueε converges
10: return programPIC
VII.3.3 Heuristic Approaches with Promotion-One Manipulations
While the approaches described above are principled in the sense that they yield provable guarantees, even the iterative approach is likely to introduce too many constraints (thereby compromising social welfare which could have been achieved). One major reason for this is that it still accounts for the full space of permutation manipulations, rather than the more salient restricted space of manipulations in which a player only promotes another to the top position in her order. We now introduce two iterative heuristic approaches which directly consider this smaller set of manipulations, albeit losing theoretical guarantees.
The intuition behind our first heuristic approach is to iteratively allow each player to promote another to the top position, check if the result yieldsstrictly highersocial welfare than current allocation, if only in this case add the corresponding constraint. The full ap- proach is shown in Algorithm 11. HerePMSW(·)denotes the assignment which maximizes social welfare (i.e., solves program (VII.2)), and sw(·) denotes the social welfare of an assignment.
Algorithm 11Heuristic 1 input:initial program (VII.2) return: Heuristicε-POIC Program
1: programPPOIC ←program (VII.2)
2: Replacing the objective with (VII.3a)
3: repeat
4: fori, and j∈Rido
5: if sw(PMSW(ij,−i))>sw(PPOIC()) and corresponding constraint has not been addedthen
6: add∑k∈Nxikuik≥ui j−ε intoPPOIC
7: end if
8: end for
9: untilthe valueε converges
10: return programPPOIC
Our second heuristic approach even further relaxes the IC constraints. In the process of constraint generation, we still compute the social welfare of the manipulated profile.
welfare of the remaining players in the current program, and check if the utility ofiand j, along with social welfare of the match among remaining players, is thereby increased; if it is, we add the constraint, since this is likely the salient manipulation. The full approach for the second heuristic is described in Algorithm 12. HereUiis the utilityireceives from ranking a player (jin this case) in the first position, and−{i,j}is the profile of all players other thaniand j.
Algorithm 12Heuristic 2 input:initial program (VII.2) return: Heuristicε-POIC Program
1: programPPOIC ←program (VII.2)
2: fori, and j∈Rido
3: ifUi+uji+sw(PPOIC(−{i,j}))>sw(PPOIC())then
4: add∑k∈Nxikuik≥ui j−ε intoPPOIC
5: end if
6: end for
7: return programPPOIC