Incentive compatibility in kidney exchange problems

Abstract

The problem of kidney exchanges shares common features with the classical problem of exchange of indivisible goods studied in the mechanism design literature, while presenting additional constraints on the size of feasible exchanges. The solution of a kidney exchange problem can be summarized in a mapping from the relevant underlying characteristics of the players (patients and their donors) to the set of matchings. The goal is to select only matchings maximizing a chosen welfare function. Since the final outcome heavily depends on the private information in possess of the players, a basic requirement in order to reach efficiency is the truthful revelation of this information. We show that for the kidney exchange problem, a class of (in principle) efficient mechanisms does not enjoy the incentive compatibility property and therefore is subject to possible manipulations made by the players in order to profit of the misrepresentation of their private information.

Appendix

1.1 Indirect mechanisms in the incomplete information setting: an impossibility result

As briefly described in Section 2, a key result in the context of mechanism design is the revelation principle. In this section we show how this can be applied in order to prove an impossibility theorem as a consequence of our examples. In the paper we have analyzed incentive compatibility of a direct mechanism, i.e. a mechanism that simply asks to each patient to report his type. In this Appendix we enlarge the set of available mechanisms, taking into account also indirect mechanisms. We therefore need some more terminology.

According to Definition 5, we consider a kidney exchange problem $K=(N,\{D_i\}_{i\in N},\mathcal{T}^n,P,w,q)$. An outcome of the problem is an admissible matching and as we have already seen in Section 5, each patient has a preference on the set of possible matchings, depending obviously on the set of types of the involved patients. We model these preferences by saying that each patient has a utility function $u_i:\mathcal{T}^n\times\mathcal{M}\rightarrow [0,+\infty)$ (clearly we consider the expected utility when we deal with $L(\mathcal{M})$ instead of $\mathcal{M})$. We assume that the utility function is common knowledge, but not the patient’s types.

Definition 6

Given a kidney exchange problem K, we call MW(K) the set of maximum weight matchings corresponding to K, and we denote by l its cardinality. The function f that associates to each kidney exchange problem the element in $L(\mathcal{M})$ assigning probability 1/l to each matching belonging to MW(K) is called maximum weight choice function.

As stated in Section 3 in a slightly different context, a direct mechanism consists of an outcome selection function $h:\mathcal{T}^n\rightarrow L(\mathcal{M})$. In order to introduce the class of indirect mechanisms, we recall the definition of a Bayesian game with consistent beliefs and Bayes-Nash equilibria (see [10]).

Definition 7

A Bayesian game is given by $(N,A, \mathbb{T},$ q,(v _i ) _i ∈ N ) where N: = {1,...,n} is a set of players, A: = ∏  _i ∈ N A _i is a collection of sets of actions and $\mathbb{T}=\prod_{i\in N} T_i$, with a probability distribution q over it, with T _i the set of types of each player i, and a utility function $v_i:\mathbb{T}\times A\rightarrow \mathbb{R}$. A pure strategy in a Bayesian game is a mapping from types to actions, i.e. s _i :T _i →A _i .

Definition 8

The profile of strategies $(s^*_1,\dots,s^*_n)$ is a Bayes-Nash equilibrium if for every agent i, for every type t _i  ∈ T _i , and every alternative strategy s _i  ∈ A _i , we have

$$\begin{array}{lll} &&\sum\limits_i q(t_{-i}|t_i)v_i\left(t,s^*_i(t_i),s^*_{-i}(t_{-i})\right)\\&&{\kern-7pt} {\kern1pc} \geq \sum\limits_i q(t_{-i}|t_i) v_i\left(t, s_i(t_i),s^*_{-i}(t_{-i})\right). \end{array}$$

(We recall that the vector v _− i is obtained from the vector v by deleting the i-th component).

Definition 9

A mechanism for the kidney exchange problem in the incomplete information case $K=(N,\{D_i\}_{i\in N},\mathcal{T}^n,P,w,q)$ is given by a triple (A,o,s) where:

• A = ∏  _i ∈ N A _i is a collection of sets of actions;

• $o:A\rightarrow L(\mathcal{M})$ is an outcome function;

• $s_i:\mathcal{T}\rightarrow A_i$ is a set of strategies.

Definition 10

A mechanism implements the maximum weight rule f for a kidney exchange problem K in Bayes-Nash equilibrium if there is a Bayes-Nash equilibrium $s^*=(s^*_1,\ldots,s^*_n)$ of the Bayesian game (N,A, $\mathcal{T}^n,q,(u_i\circ o)_{i\in N} )$ such that for all $(t_1,\ldots,t_n)\in \mathcal{T}^n$ it holds $o(s^*(t_1,\ldots,t_n))=f(t_1,\ldots,t_n)$.

We recall the fundamental result that we have cited several times, the so called revelation principle, restricted to our case (see [1], Ch. 7, p.174).

Theorem 1

Suppose that there is a mechanism that implements the maximum weight social choice rule f in Bayes-Nash equilibrium. Then there exists a Bayes-Nash equilibrium incentive compatible direct mechanism that also implements f in Bayes-Nash equilibrium (with the truth-telling equilibrium).

In the rest of the paper we have shown that there exist some cases in which the direct mechanism does not truthfully implement the maximum weight social choice function. More precisely, in Example 2, we have studied such a case. As a consequence of the revelation principle, this implies that for the problem in Example 2 the maximum weight social choice rule is not Bayes-Nash implementable through any mechanism. More formally, we can collect these considerations in the following theorem.

Theorem 2

When there are at least three patients, there exist a set of possible types $\mathcal{T}$ and a probability distribution on $\mathcal{T}$ such that no mechanism exists that Bayes-Nash implements the maximum weight social rule for every realization of the kidney exchange problem.