Full implementation of social choice functions in dominant strategies

Abstract

We consider the classical mechanism design problem of fully implementing social choice functions in dominant strategies in settings where monetary payments are allowed and the utility functions are quasi-linear. We consider both the general question of full implementation by indirect mechanisms and the special case of full implementation by incentive compatible direct revelation mechanisms. For the general case of full implementation by indirect mechanisms, we prove that one can restrict attention to incentive compatible augmented revelation mechanisms, in which the type space of each agent is a subset of the set of her possible bids and truthful reporting is a dominant strategy equilibrium. When the type spaces of the agents are finite, we give a complete characterization of the set of social choice functions that can be fully implemented in dominant strategies. For the case that one restricts to incentive compatible direct revelation mechanisms, we show that an adaption of the well-known negative cycle criterion for partial implementability also characterizes the social choice functions that are fully implementable.

Appendices

Appendix A

Here, we present the reformulation of the system given by Inequalities (2) and (3) that is used in order to prove Theorem 4.

In the first step, we put the valuations to the right-hand side of each Inequality (2) and (3) and the payments to the left-hand side. Writing the resulting difference

$$\begin{aligned} V_i(f(\theta _i,\theta _{-i}^{\prime }),\theta )-V_i(f(\theta _i^{\prime },\theta _{-i}^{\prime }),\theta ) \end{aligned}$$

of the two valuations of agent i on the right-hand sides as $c_i^{(\theta _{-i},\theta _{-i}^{\prime })}(\theta _i,\theta _i^{\prime })$, the system becomes

Observe that the left-hand sides $P_i(\theta _i^{\prime },\theta _{-i}^{\prime }) - P_i(\theta _i,\theta _{-i}^{\prime })$ of the Inequalities (4) and (5) are now independent of the type vector $\theta _{-i}$ of all agents except i. Thus, the payment difference $P_i(\theta _i^{\prime },\theta _{-i}^{\prime }) - P_i(\theta _i,\theta _{-i}^{\prime })$ will be less or equal to the value $c_i^{(\theta _{-i},\theta _{-i}^{\prime })}(\theta _i,\theta _i^{\prime })$ on the right-hand side of (4) for all $\theta _{-i}$ if and only it is less or equal to the infimum of all these values. Analogously, it will be strictly smaller than one of the right-hand sides of (5) if and only if it is strictly smaller than the supremum of all of these. Hence, we define

$$\begin{aligned} \underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime }):= & {} \inf _{\theta _{-i}\in \varTheta _{-i}} c_i^{(\theta _{-i},\theta _{-i}^{\prime })}(\theta _i,\theta _i^{\prime })\\= & {} \inf _{\theta _{-i}\in \varTheta _{-i}} \biggl (V_i(f(\theta _i,\theta _{-i}^{\prime }),\theta )-V_i(f(\theta _i^{\prime },\theta _{-i}^{\prime }),\theta )\biggr ) \end{aligned}$$

and

$$\begin{aligned} {\bar{c}}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime }):= & {} \sup _{\theta _{-i}\in \varTheta _{-i}} c_i^{(\theta _{-i},\theta _{-i}^{\prime })}(\theta _i,\theta _i^{\prime })\\= & {} \sup _{\theta _{-i}\in \varTheta _{-i}} \biggl (V_i(f(\theta _i,\theta _{-i}^{\prime }),\theta )-V_i(f(\theta _i^{\prime },\theta _{-i}^{\prime }),\theta )\biggr ) \end{aligned}$$

for $\theta _i,\theta _i^{\prime }\in \varTheta _i$, and rewrite the system as

Note that, in the case that some value $\underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })$ is equal to $-\infty$, it follows that the system does not have a solution and f cannot be implemented at all (not even partially). Hence, we assume from now on that all values $\underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })$ are finite. On the other hand, the values ${\bar{c}}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })$ defined by the suprema may be infinite without contradicting the implementability of f.

Moreover, observe that, whenever $\underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })< {\bar{c}}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })$ for some $\theta _{-i}^{\prime }\in \varTheta _{-i}$, the second condition follows automatically from the first one for this pair $(\theta _i,\theta _i^{\prime })$. Hence, we only have to include the second condition in the case that the two values are the same, and, in this case, we can replace ${\bar{c}}$ by $\underline{c}$ in the second condition:

Now we show that the payment $P_i(\theta _i,\theta _{-i}^{\prime })$ does in fact only depend on $\theta _i$ through the outcome $f(\theta _i,\theta _{-i}^{\prime })$ chosen when agent i bids $\theta _i$. To this end, we fix an agent i and a vector $\theta _{-i}^{\prime }\in \varTheta _{-i}$ of bids of the other agents and consider a pair $(\theta _i,\theta _i^{\prime })\in \varTheta _i^2$ of types of agent i such that $f(\theta _i,\theta _{-i}^{\prime })=f(\theta _i^{\prime },\theta _{-i}^{\prime })=:x\in X$. Then we have

$$\begin{aligned} \underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })= & {} \inf _{\theta _{-i}\in \varTheta _{-i}} \biggl (V_i(f(\theta _i,\theta _{-i}^{\prime }),\theta )-V_i(f(\theta _i^{\prime },\theta _{-i}^{\prime }),\theta )\biggr )\\= & {} \inf _{\theta _{-i}\in \varTheta _{-i}} \biggl (V_i(x,\theta )-V_i(x,\theta )\biggr ) = 0 \end{aligned}$$

and analogously $\underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i^{\prime },\theta _i)=0$. Hence, the Inequalities (8) corresponding to $(\theta _i,\theta _i^{\prime })$ imply that

$$\begin{aligned} P_i(\theta _i^{\prime },\theta _{-i}^{\prime }) - P_i(\theta _i,\theta _{-i}^{\prime })\le & {} 0 \quad \text {and}\quad P_i(\theta _i,\theta _{-i}^{\prime }) - P_i(\theta _i^{\prime },\theta _{-i}^{\prime }) \le 0, \end{aligned}$$

which yields $P_i(\theta _i^{\prime },\theta _{-i}^{\prime })=P_i(\theta _i,\theta _{-i}^{\prime })$. Hence, for every $x\in X$ that results as the outcome $f(\theta _i,\theta _{-i}^{\prime })$ for some $\theta _i\in \varTheta _i$, we can define

$$\begin{aligned} P_i^{\theta _{-i}^{\prime }}(x):=P_i(\theta _i,\theta _{-i}^{\prime })\quad \text { for some } \theta _i\in \varTheta _i \text { with } f(\theta _i,\theta _{-i}^{\prime })=x. \end{aligned}$$

Furthermore, for each pair $(\theta _i,\theta _i^{\prime })$ of bids of agent i, we denote the set of all bid vectors $\theta _{-i}^{\prime }$ of the other agents such that the outcome when agent i bids $\theta _i$ differs from the outcome when she bids $\theta _i^{\prime }$ by $C(\theta _i,\theta _i^{\prime })$:

$$\begin{aligned} C(\theta _i,\theta _i^{\prime }):=\{\theta _{-i}^{\prime }\in \varTheta _{-i}:f(\theta _i,\theta _{-i}^{\prime }) \ne f(\theta _i^{\prime },\theta _{-i}^{\prime })\}, \end{aligned}$$

Thus, the system translates to

As observed above, the left-hand sides of Inequalities (10) and (11) are now independent of $\theta _i$ and $\theta _i^{\prime }$ as long as the respective outcomes $f(\theta _i,\theta _{-i}^{\prime })$ and $f(\theta _i^{\prime },\theta _{-i}^{\prime })$ do not change. Hence, we may replace all Inequalities (10) with the same outcomes appearing on the left-hand side by a single inequality in which the right-hand side is replaced by the infimum. Denoting the set of all outcomes $x\in X$ that are chosen by f for some bid of agent i when the others bid $\theta _{-i}^{\prime }$ by

$$\begin{aligned} W(\theta _{-i}^{\prime }):=\{x\in X:\exists \theta _i\in \varTheta _i: f(\theta _i,\theta _{-i}^{\prime })=x\}, \end{aligned}$$

we, thus, define

$$\begin{aligned} c_i^{\theta _{-i}^{\prime }}(x,x^{\prime }):=\inf _{\begin{array}{ll} \scriptstyle \theta _i,\theta _i^{\prime }\in \varTheta _i:\\ \scriptstyle f(\theta _i,\theta _{-i}^{\prime })=x\\ \scriptstyle f(\theta _i^{\prime },\theta _{-i}^{\prime })=x^{\prime } \end{array}} \underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime }) > -\infty \end{aligned}$$

for all $x,x^{\prime }\in W(\theta _{-i}^{\prime })$. Furthermore, we introduce the notation

$$\begin{aligned} K^{\theta _{-i}^{\prime }}(x):=\{\theta _i\in \varTheta _i:f(\theta _i,\theta _{-i}^{\prime })=x\} \end{aligned}$$

for the preimage under f of an outcome $x\in X$ when the other agents bid $\theta _{-i}^{\prime }$. The system then translates to

Now observe that, whenever $\theta _i\in K^{\theta _{-i}^{\prime }}(x),\theta _i^{\prime }\in K^{\theta _{-i}^{\prime }}(x^{\prime })$, and $c_i^{\theta _{-i}^{\prime }}(x,x^{\prime })< \underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })$ for some$\theta _{-i}^{\prime }\in \varTheta _{-i}$, the second condition follows automatically from the first one for this pair $(\theta _i,\theta _i^{\prime })$. Hence, we just have to consider the second condition for the pairs $(\theta _i,\theta _i^{\prime })$ for which $c_i^{\theta _{-i}^{\prime }}(x,x^{\prime })=\underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime })$ for all$\theta _{-i}^{\prime }\in \varTheta _{-i}$, $x,x^{\prime }$ with $\theta _i\in K^{\theta _{-i}^{\prime }}(x)$ and $\theta _i^{\prime }\in K^{\theta _{-i}^{\prime }}(x^{\prime })$. Thus, the system can be rewritten as

Finally, the second condition in the system can be reformulated as follows: All conditions on the pairs $(\theta _i,\theta _i^{\prime })$ do in fact only depend on the second value $\theta _i^{\prime }$ through the corresponding outcomes $f(\theta _i^{\prime },\theta _{-i}^{\prime })$. In particular, the values

$$\begin{aligned} \underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,x^{\prime }):= & {} \underline{c}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime }) \text { for some } \theta _i^{\prime }\in K^{\theta _{-i}^{\prime }}(x^{\prime }) \\ {\bar{c}}_i^{\theta _{-i}^{\prime }}(\theta _i,x^{\prime }):= & {} {\bar{c}}_i^{\theta _{-i}^{\prime }}(\theta _i,\theta _i^{\prime }) \text { for some } \theta _i^{\prime }\in K^{\theta _{-i}^{\prime }}(x^{\prime }) \end{aligned}$$

are well-defined for all $x\in W(\theta _{-i}^{\prime })$. Defining

$$\begin{aligned} C(\theta _i,x^{\prime }):=\{\theta _{-i}^{\prime }\in \varTheta _{-i}:x^{\prime }\in W(\theta _{-i}^{\prime }), f(\theta _i,\theta _{-i}^{\prime }) \ne x^{\prime }\}, \end{aligned}$$

we can, hence, rewrite the system as

For every agent i and every fixed vector $\theta _{-i}^{\prime }\in \varTheta _{-i}$ of bids of the other agents, the Inequalities (16) corresponding to i and $\theta _{-i}^{\prime }$ are exactly equivalent to the values $P_i^{\theta _{-i}^{\prime }}(x)$ defining a node potential in the complete, directed graph $G_i(\theta _{-i}^{\prime })$ on the set $W(\theta _{-i}^{\prime })$ with the cost of the arc from outcome x to $x^{\prime }$ given as $c_i^{\theta _{-i}^{\prime }}(x,x^{\prime })$.

The Inequalities (17) state that some arcs in the graphs must have strictly positive reduced cost: For each agent i, each outcome $x^{\prime }$, and each type $\theta _i$ of agent i, consider the graphs $G_i(\theta _{-i}^{\prime })$ in which $x^{\prime }$ is contained as a node, but is not the outcome resulting from truthful reporting by i (i.e., $x^{\prime }\ne f(\theta _i,\theta _{-i}^{\prime })$). Whenever (for some fixed i, $x^{\prime }$, $\theta _i$) it holds for all of these graphs that the value $V_i(f(\theta _i,\theta _{-i}^{\prime }),\theta )-V_i(x^{\prime },\theta )$ used in the calculation of the arc costs is independent of the type vector $\theta _{-i}$ of the other agents and independent of the true type $\theta _i$ of agent i as long as the outcome $f(\theta _i,\theta _{-i}^{\prime })$ resulting from truthful reporting by i does not change, then there exists at least one graph $G_i(\theta _{-i}^{\prime })$ in which the arc $(f(\theta _i,\theta _{-i}^{\prime }),x^{\prime })$ has strictly positive reduced cost. Hence, we obtain exactly the characterization of social choice functions that can be (fully) implemented by incentive compatible direct revelation mechanisms stated in Theorem 4.

Appendix B

Here, we show that, in the case where that the type spaces $\varTheta _i$ of the agents are finite, we can use the characterization obtained in Theorem 4 to efficiently compute the payments of an incentive compatible direct revelation mechanism that fully implements the social choice function f in dominant strategies or decide that none exist.

We first compute the shortest path distances from an arbitrary node ${\bar{x}}$ to all other nodes in the graph $G_i(\theta _{-i}^{\prime })$ for every $i\in N$ and every $\theta _{-i}^{\prime }\in \varTheta _{-i}$ (which can be done efficiently using the Bellman-Ford Algorithm, cf. Cormen et al. (2009)). In the case that one of the graphs $G_i(\theta _{-i}^{\prime })$ contains a negative cycle (so no node potentials exist in this graph), Theorem 4 shows that the given social choice function f cannot be implemented at all (not even partially). Otherwise, denoting the computed shortest path distance of a node x in the graph $G_i(\theta _{-i}^{\prime })$ by $P_i^{\theta _{-i}^{\prime }}(x)$, we obtain payments partially implementing the social choice function f by setting

$$\begin{aligned} P_i(\theta ^{\prime }):=P_i^{\theta _{-i}^{\prime }}(f(\theta ^{\prime })) \quad \text {for} \quad i=1,\dots ,n \quad \text {and} \quad \theta ^{\prime }\in \varTheta . \end{aligned}$$

For full implementation, we have to modify the node potentials obtained by shortest path computations such that some arcs in the graphs $G_i(\theta _{-i}^{\prime })$ have strictly positive reduced cost. This is done via the following procedure: For every agent i and every $\theta _{-i}^{\prime }\in \varTheta _{-i}$, we again consider the graph $G_i(\theta _{-i}^{\prime })$ and the corresponding node potentials $P_i^{\theta _{-i}^{\prime }}(x)$ for $x\in W(\theta _{-i}^{\prime })$. We delete all arcs $(x,x^{\prime })$ that already have strictly positive reduced cost from $G_i(\theta _{-i}^{\prime })$. After doing so, we also delete all isolated nodes. In the remaining graph, which contains only arcs with reduced cost zero, we then search for a node x with no outgoing arcs. For such a node x, the minimum of the reduced costs of its outgoing arcs given by

$$\begin{aligned} \epsilon (x):=\min _{x^{\prime }\in W(\theta _{-i}^{\prime })} \biggl (c_i^{\theta _{-i}^{\prime }}(x,x^{\prime }) - P_i^{\theta _{-i}^{\prime }}(x^{\prime }) + P_i^{\theta _{-i}^{\prime }}(x)\biggr ) \end{aligned}$$

is strictly positive. Thus, we can lower $P_i^{\theta _{-i}^{\prime }}(x)$ by $\epsilon (x)/2$ without making the reduced cost of any arc become negative. After doing so, all arcs with end node x have strictly positive reduced cost, so we can delete these arcs and the node x from the graph $G_i(\theta _{-i}^{\prime })$.

To find a node x in $G_i(\theta _{-i}^{\prime })$ with no outgoing arcs, we use depth-first search (DFS) starting with an arbitrary node in the graph. Doing so, we either find a node with no outgoing arcs, or we discover a directed cycle. In the first case, we lower the potential of the node as in the procedure described above and continue the DFS-procedure at the node considered before as long as there are still nodes remaining in the graph. In the case that we find a directed cycle C, all arcs on C have reduced cost zero, which means that

$$\begin{aligned} 0=\sum _{(x_k,x_l)\in C}\left( P_i^{\theta _{-i}^{\prime }}(x_l)-P_i^{\theta _{-i}^{\prime }}(x_k)\right) = \sum _{(x_k,x_l)\in C} c_i^{\theta _{-i}^{\prime }}(x_k,x_l), \end{aligned}$$

(18)

where the first equality follows since C is a cycle.

On the other hand, if any arc on C had strictly positive reduced cost, we would obtain

$$\begin{aligned} 0=\sum _{(x_k,x_l)\in C}\left( P_i^{\theta _{-i}^{\prime }}(x_l)-P_i^{\theta _{-i}^{\prime }}(x_k)\right) < \sum _{(x_k,x_l)\in C} c_i^{\theta _{-i}^{\prime }}(x_k,x_l), \end{aligned}$$

contradicting (18). Hence, the reduced cost cannot be made strictly positive for any arc on the cycle C. In this case, we contract all nodes on C to a single supernode and continue the DFS-procedure at this new node (see Fig. 1). The new supernode does not receive a node potential. Instead, the node potentials of the original nodes corresponding to the supernode are used in order to calculate the reduced costs of arcs incident to the supernode in all further steps of the algorithm. When the algorithm would have to lower the potential of the supernode later, we lower the potentials of all original nodes corresponding to the supernode by the same amount, which preserves the validity of (18).

Fig. 1

Contraction of a cycle to a supernode

When the above procedure terminates, every arc $(x,x^{\prime })$ in the original graph $G_i(\theta _{-i}^{\prime })$ either has strictly positive reduced cost, or is contained in a cycle of arcs with reduced cost zero (so the reduced cost of $(x,x^{\prime })$ cannot be made strictly positive in this graph). Hence, the condition for full implementation from Theorem 4 is either

• (i) fulfilled for the node potentials obtained after applying the procedure to all graphs $G_i(\theta _{-i}^{\prime })$, so the payments corresponding to these node potentials can be used to fully implement the given social choice function f, or

• (ii) violated, so the social choice function f cannot be fully implemented by an incentive compatible direct revelation mechanism.

Using the known running time bounds for the Bellman-Ford Algorithm and DFS (cf. Cormen et al. (2009)), the running time of the complete procedure described above can easily be shown to be in ${\mathcal {O}}(n\cdot |\varTheta |\cdot |X|^3)$. Hence, we obtain the following theorem:

Theorem 5

When the type spaces $\varTheta _i$ of the agents are finite, the above procedure correctly computes the payments P of an incentive compatible direct revelation mechanism $\varGamma _{(f,P)}$ that fully implements the given social choice function f in dominant strategies or decides that no such payments exist. The procedure runs in time ${\mathcal {O}}(n\cdot |\varTheta |\cdot |X|^3)$.