Implementation in stochastic dominance Nash equilibria

Abstract

We study solutions that choose lotteries for profiles of preferences defined over sure alternatives. We define Nash equilibria based on “stochastic dominance” comparisons and study the implementability of solutions in such equilibria. We show that a Maskin-style invariance condition is necessary and sufficient for implementability. Our results apply to an abstract Arrovian environment as well as a broad class of economic environments.

Appendices

A The inclusion relations of Proposition 1 may be strict

For each $P\in \mathcal {P}^N$ and each $a\in A$, let N(a, P) be the set of agents who most prefer a. That is, $N(a,P) \equiv \{i\in N: a~P_i~b$ for each $b\in A{\setminus }\{a\}\}$. Let M(p) be the set plurality-winners: Each is most preferred by at least as many agents as any other alternative. That is, $M(P)\equiv \{a\in A: |N(a,P)|\ge |N(b,P)|$ for each $b\in A\}$.

Let $\varphi$ be the single valued solution that picks each plurality winner with the same probability. That is, for each $P\in \mathcal {P}^N$,

$$\begin{aligned} \varphi (P)(a)\equiv \left\{ \begin{array}{cl} \frac{1}{|M(P)|}, &{} \quad \text {if }\, a\in M(P),\\ 0, &{} \quad \text {otherwise.}\end{array}\right. \end{aligned}$$

Let $N\equiv \{1,2,3\}$, $A\equiv \{a,b,c\}$, $P\in \mathcal {P}^N$, and $u\in \mathcal {U}(P)$ be such that

$$\begin{aligned} \begin{array}{ccc}P_1&{} P_2 &{} P_3\\ \hline a &{} b&{} c\\ b&{} c&{} a\\ c&{} a&{} b\end{array} \qquad \begin{array}{lll} u_1(a)=100 &{} \quad u_2(a)=0 &{} \quad u_3(a)=99\\ u_1(b)=1 &{} \quad u_2(b)=100 &{} \quad u_3(b)=0\\ u_1(c)=0 &{}\quad u_2(c)=1 &{} \quad u_3(c)=100\end{array} \end{aligned}$$

Consider the preference revelation game induced by $\varphi$, $\Gamma \equiv (\mathcal {P}^N, \varphi )$. It is easy to check that for each $i\in N$, $(P_i,P_i,P_i)\in SNE^{sd}(\Gamma ,P)$ since the outcome does not change no matter how an agent deviates.

Next, we show that $(P_1,P_2,P_1)\in NE(\Gamma ,u){\setminus } SNE^{sd}(\Gamma ,P)$. Note that $\varphi (P_1,P_2,P_1)=(1,0,0)$ and if agent 3 reports $P_3$, then $\varphi (P_1,P_2,P_3)=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. Therefore, $(P_1,P_2,P_1)\notin SNE^{sd}(\Gamma ,P)$. We now show that $(P_1,P_2,P_1)\in NE(\Gamma ,u)$. First, the outcome does not change no matter how agent 2 deviates. Second, agent 1 does not have an incentive to deviate from $(P_1,P_2,P_1)$ because $\varphi (P_1,P_2,P_1)=(1,0,0)$ is his most preferred outcome at $P_1$. Third, agent 3 does not have an incentive to report b as his favorite alternative since the resulting outcome (0, 1, 0) is his least preferred outcome at $P_3$. Agent 3 does not have an incentive to report c as his favorite alternative either since the outcome changes from (1, 0, 0) to $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ and his expected utility decreases.

Lastly, we show that $(P_2,P_2,P_3)\in WNE^{sd}(\Gamma ,P){\setminus } NE(\Gamma , u)$. Note that $\varphi (P_2,P_2,P_3)=(0,1,0)$ and if agent 1 reports $P_1$, then $\varphi (P_1,P_2,P_3)=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$. It is easy to check that agent 1’s expected utility increases by doing so. Therefore, $(P_2,P_2,P_3)\notin NE(\Gamma ,u)$. We now show that $(P_2,P_2,P_3)\in WNE(\Gamma , P)$. First, the outcome does not change no matter how agent 3 deviates. Second, agent 2 does not have an incentive to deviate because $\varphi (P_2,P_2,P_3)=(0,1,0)$ is his most preferred outcome at $P_2$. Third, agent 1 does not have an incentive to report c as his favorite alternative since the resulting outcome (0, 0, 1) is his least preferred outcome at $P_1$. If agent 1 reports a as his favorite alternative, then the resulting outcome is $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ which does not stochastically dominate (0, 1, 0) at $P_1$. Therefore, $(P_2,P_2,P_3)\in WNE(\Gamma ,P)$.

B Proof of Proposition 2

We prove part (1) of Proposition 2: a solution $\varphi$ is implementable in strong SDNE if and only if it is invariant to lower SD monotonic transformations. The proofs of parts (2) and (3) are very similar. We omit them but highlight the differences at the end.

(${\varvec{\Rightarrow }}$) Suppose that $\varphi$ is implementable in strong SDNE. Suppose that it is implemented by $\Gamma \equiv (S,h)$. Let $P\in \mathcal {P}^N$ and $\pi \in \varphi (P)$. Then, there is $s\in SNE^{sd}(\Gamma , P)$ such that $h(s)=\pi$. Since $s\in SNE^{sd}(\Gamma , P)$, for each $i\in N$ and each $s_i'\in S_i$, $h(s_i',s_{-i})\in L^{sd}(P_i,\pi )$.

Let $P'\in \mathcal {P}^N$ be such that for each $i\in N$, $L^{sd}(P_i',\pi )\supseteq L^{sd}(P_i,\pi )$. Then, for each $s_i'\in S_i$, $h(s_i',s_{-i})\in L^{sd}(P_i',\pi )$. Thus, $s\in SNE^{sd}(\Gamma ,P')$ so $\pi =h(s) \in \varphi (P')$.

(${\varvec{\Leftarrow }}$) In this part of the proof we define a game form and show that it implements $\varphi$ in strong SDNE.

First, we present some notation. For each $P_0\in \mathcal {P}$ and each $B\subseteq A$, let $\tau (P_0,B)$ be the top alternative under ${\varvec{P}}_{\varvec{0}}$ in ${\varvec{B}}$. That is, $\tau (P_0,B)=a$ if and only if $a\in B$ and for each $b\in B {\setminus }\{a\}$, $a \,P_0 b$. For convenience, we usually drop A in the expression $\tau (P_0,A)$ and write $\tau (P_0)$ instead. That is, when the second argument is omitted, it is A. For each $a\in A$, let ${\varvec{\delta }}^{\varvec{a}}$ $\in \Delta (A)$ be such that $\delta ^a(a) = 1$. Let ${\varvec{D}}$ $\equiv \{\delta ^a:a\in A\}$. Let $\bar{{\varvec{\pi }}}$ $\in \Delta (A)$ be such that for each $a\in A$, $\bar{\pi }(a)\equiv \frac{1}{|A|}$.

Next, we define a canonical game form. For each $i\in N,$ let $S_i \equiv \mathcal {P}^N\times \Delta (A)\times \Delta (A) \times {\mathbb {N}}_+.$ Denote a generic member of $S_i$ by $s_i\equiv (P^i,\pi ^i,\sigma ^i,n^i)$. Define $h:S\rightarrow \Delta (A)$ by setting, for each $s\in S$,

$$\begin{aligned} h(s) \equiv \left\{ \begin{array}{ll} \pi &{}\quad \text {if for each}\, i\in N, s_i = (P,\pi ,\sigma ,n)\, \text {and}\, \pi \in \varphi (P)\\ \\ \sigma ^i&{} \quad \left\{ \begin{array}{l}\text {if there is }i\in N\text { such that} \\ \text {for each }j\in N {\setminus } \{i\}, s_j = (P,\pi ,\sigma ,n) \ne s_i\\ \quad =(P^i,\pi ^i,\sigma ^i,n^i) \\ \text {and } \sigma ^i\in L^{sd}(P_i,\pi ){\setminus } D,^{21} \end{array}\right. \\ \\ \pi &{} \quad \left\{ \begin{array}{l}\text {if there is }i\in N\text { such that} \\ \text {for each }j\in N {\setminus } \{i\}, s_j = (P,\pi ,\sigma ,n) \ne s_i\\ \quad =(P^i,\pi ^i,\sigma ^i,n^i) \\ \text {and } \sigma ^i\not \in L^{sd}(P_i,\pi ){\setminus } D,\end{array}\right. \\ \\ \frac{1}{n^{i^*}+1}\sigma ^{i^*}+\frac{n^{i^*}}{n^{i^*}+1}\pi ^{i^*} &{}\quad \left\{ \begin{array}{l}\text {if there is a triple}\, i,j,k\in N \,\text {such that} s_i\ne s_j,\\ \quad s_i\ne s_k,s_j \ne s_k,\\ \text {and}\, \pi ^{i^*}\ne \sigma ^{i^*},\, \text {where} i^*\equiv \min \left\{ \text {arg}\max \limits _{j\in N} n^j\right\} ,\\ \quad \text {and }\end{array}\right. \\ \\ \bar{\pi } &{} \quad \text {otherwise.} \end{array} \right. \end{aligned}$$

²¹

Let $\Gamma \equiv (S,h)$.²² Our goal is to show that for each $P\in \mathcal {P}^N$, $h(SNE^{sd}(\Gamma , P)) = \varphi (P)$.

First, we show that $\varphi (P) \subseteq h(SNE^{sd}(\Gamma , P))$. Let $\pi \in \varphi (P)$. Let $s\in S$ be such that for each $i\in N$, $s_i = (P, \pi , \pi ,0 )$. For each $i\in N$ and each $s_i'\in S_i{\setminus } \{s_i\}$, either $h(s_i',s_{-i}) = h(s) = \pi$ or $h(s_i',s_{-i}) \in L^{sd}(P_i,\pi ){\setminus } D.$ Thus, $s\in SNE^{sd}(\Gamma , P)$. Consequently $\pi =h(s) \in h(SNE^{sd}(\Gamma , P))$.

Next, consider $\overset{\tiny \text {T}}{P}\in \mathcal {P}^N$ and $s\in SNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. We complete the proof by showing that $h(s)\in \varphi (\overset{\tiny \text {T}}{P})$. Let $\pi \equiv h(s)$.

Recall that for each $P_0\in \mathcal {P}$ and each $\pi \in \Delta (A)$, $L^{sd}(P_0,\pi )$ is closed and convex. Thus, for each $P_0'\in \mathcal {P}$ if $L^{sd}(P_0',\pi )\supseteq L^{sd}(P_0,\pi ){\setminus } D$ then $L^{sd}(P_0',\pi )\supseteq L^{sd}(P_0,\pi ).$ ²³

Case 1: For each $i\in N$, $s_i = (P,\pi ,\sigma , n)$ and $\pi \in \varphi (P)$. Then, for each $i\in N$, $L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )\supseteq L^{sd}(P_i,\pi ){\setminus } D$: Otherwise, there is $\sigma ^i\in \Delta (A)$ such that $\sigma ^i \in L^{sd}(P_i,\pi ) {\setminus } (D\cup L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )).$ Let $s_i'\equiv (P,\pi ,\sigma ^i,n)$. Then $h(s_i',s_{-i}) = \sigma ^i$. So $h(s_i',s_{-i}) = \sigma ^i\notin L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ) = L^{sd}(\overset{\tiny \text {T}}{P}_i,h(s))$. This contradicts $s\in SNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. Since $\pi \in \varphi (P)$ and for each $i\in N$, $L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )\supseteq L^{sd}(P_i,\pi )$, by invariance to lower SD monotonic transformations, $\pi \in \varphi (\overset{\tiny \text {T}}{P})$.

Case 2: There is $i\in N$ such that, for each $j\in N{\setminus }\{i\}, s_j = (P,\pi ,\sigma , n)\ne s_i$. Let $\tilde{\pi }\equiv h(s)$. Then for each $j\in N{\setminus }\{i\}$, $\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{P}_j)}$: otherwise, let $\hat{s}_j\in S_j$ be such that ${\hat{n}^j} \equiv \max \{n^i,n\}+1$, ${\hat{\sigma }^j} \equiv \tilde{\pi }$, ${\hat{P}^j}\in \mathcal {P}^N$, and ${\hat{\pi }^j} = \delta ^{\tau (\overset{\tiny \text {T}}{P_j})}$. Then, $h(\hat{s}_j,s_{-j}) = \frac{1}{{\hat{n}^j}+1}\tilde{\pi }+ \frac{{\hat{n}^j}}{{\hat{n}^j}+1} \delta ^{\tau (\overset{\tiny \text {T}}{P_j})} \,{\overset{\tiny \text {T}}{P_j^{sd}}} \tilde{\pi }$. This contradicts $s\in SNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. Thus, for each $j\in N{\setminus }\{i\}$, $\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{P}_j)}$.²⁴

Since $\tilde{\pi }\in D$, $\tilde{\pi }\ne \sigma ^i$ so, $\tilde{\pi }= \pi$. This implies that $L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )\supseteq L^{sd}(P_i,\pi ){\setminus } D$: Otherwise, there is $\sigma ^i\in \Delta (A)$ such that $\sigma ^i\in L^{sd}(P_i,\pi ) {\setminus } (D\cup L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ))$. Let $s_i'\equiv (P,\pi ,\sigma ^i,n)$. Then, $h(s_i',s_{-i}) = \sigma ^i$. So $h(s_i',s_{-i}) = \sigma ^i \notin L^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ) = L^{sd}(\overset{\tiny \text {T}}{P}_i, h(s))$. This contradicts $s\in SNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. Since $\pi \in \varphi (P)$ and for each $k\in N$, $L^{sd}(\overset{\tiny \text {T}}{P}_k,\pi )\supseteq L^{sd}(P_k,\pi )$, by invariance to lower SD monotonic transformations, $\pi \in \varphi (\overset{\tiny \text {T}}{P})$.

Case 3: There is a triple $i,j,k\in N$ such that $s_i \ne s_j, s_i\ne s_k,$ and $s_j\ne s_k$. Recall that $\pi \equiv h(s)$.

Fig. 2

Illustration of the proof of Proposition 2. For each sub-case of 3.1, we see that $h(\hat{s}_{i^*}, s_{-i^*}) \,{P^{sd}_{i^*}} h(s)$: a corresponds to the case where $\pi ^{i^*} \,{{P^{sd}_{i^*}}} \sigma ^{i^*}$; b corresponds to $\sigma ^{i^*} \,{{P^{sd}_{i^*}}} \pi ^{i^*}$; and c corresponds to the remaining case

Case 3.1: $\pi = \frac{1}{n^{i^*}+1}\sigma ^{i^*}+\frac{n^{i^*}}{n^{i^*}+1}\pi ^{i^*}$ where $i^*\equiv \min \left\{ \text {arg}\max \limits _{j\in N} n^j\right\}$.

(Figure 2a) If $\pi ^{i^*}\,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \sigma ^{i^*}$, then let $\hat{s}_{i^*} \equiv (P^{i^*},\pi ^{i^*},\sigma ^{i^*},n^{i^*}+1)$.

(Figure 2b) If $\sigma ^{i^*}\,{\overset{\tiny \text {T}}{P^{sd}}_{i^*}} \pi ^{i^*}$, then let $\hat{s}_{i^*} \equiv (P^{i^*},\sigma ^{i^*},\pi ^{i^*},n^{i^*}+1)$. Note that $\sigma ^{i^*}$ and $\pi ^{i^*}$ are interchanged.

(Figure 2c) Otherwise, let $\hat{s}_{i^*} \equiv (P^{i^*}, \hat{\pi }^{i^*}, \sigma ^{i^*},\hat{n}^{i^*})$, where $\hat{\pi }^{i^*} = \delta ^{\tau (\overset{\tiny \text {T}}{P}_{i^*})}$ and $\hat{n}^{i^*}$ is large enough that $\hat{n}^{i^*} > n^{i^*}$ and $\frac{1}{\hat{n}^{i^*}+1}\sigma ^{i^*}+\frac{\hat{n}^{i^*}}{\hat{n}^{i^*}+1}\hat{\pi }^{i^*} \,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \pi$.

For each of the above cases, $h(\hat{s}_{i^*}, s_{-i^*}) \,{\overset{\tiny \text {T}}{P^{sd}_{i^*}}} \pi = h(s)$. This contradicts $s\in SNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$.

Case 3.2: $\pi = \bar{\pi }$. Let $\hat{s}_{i^*} \equiv (\hat{P}^{i^*},\hat{\pi }^{i^*},\hat{\sigma }^{i^*},\hat{n}^{i^*}) \in S_{i^*}$ be such that $\hat{\pi }^{i^*} = \delta ^{\tau (P_{i^*})}$, $\hat{\sigma }^{i^*}(\ne \hat{\pi }^{i^*}) \,{\overset{\tiny \text {T}}{ P_{i^*}^{sd}}} \bar{\pi }$, and $\hat{n}^{i^*} \equiv {\max _{l\in N} n^l}+2$. Then, $h(\hat{s}_{i^*}, s_{-i^*}) = \frac{1}{\hat{n}^{i^*}}\hat{\sigma }^{i^*}+\frac{\hat{n}^{i^*}}{\hat{n}^{i^*}+1}\hat{\pi }^{i^*} \,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \bar{\pi }= h(s)$. This contradicts $s\in SNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$.

We omit the proofs of parts (2) and (3) of Proposition 2. To prove (2), we simply adapt the proof of (1) by replacing “$\sigma ^i\in L^{sd}(P_i,\pi ){\setminus } D$” in the definition of h with “$\sigma ^i\notin U^{sd}(P_i,\pi )\cup D$.” To prove (3), we replace it with “$\sigma ^i\in L^{eu}(u_i,\pi ){\setminus } D$.”

1.1 B.1 The proofs of (2) and (3) of Proposition 2 for referees. Not for publication

We present the complete proofs with the changes mentioned above.

Implementation in weak SDNE

(${\varvec{\Rightarrow }}$) Suppose that $\varphi$ is implementable in weak SDNE. Suppose that it is implemented by $\Gamma \equiv (S,h)$. Let $P\in \mathcal {P}^N$ and $\pi \in \varphi (P)$. Then, there is $s\in WNE^{sd}(\Gamma , P)$ such that $h(s)=\pi$. Since $s\in WNE^{sd}(\Gamma , P)$, for each $i\in N$ and each $s_i'\in S_i$, $h(s_i',s_{-i})\notin U^{sd}(P_i,\pi ){\setminus }\{\pi \}$.

Let $P'\in \mathcal {P}^N$ be such that for each $i\in N$, $U^{sd}(P_i',\pi )\subseteq U^{sd}(P_i,\pi )$. Then, for each $s_i'\in S_i$, $h(s_i',s_{-i})\notin U^{sd}(P_i',\pi ){\setminus }\{\pi \}$. Thus, $s\in WNE^{sd}(\Gamma ,P')$ so $\pi =h(s) \in \varphi (P')$.

(${\varvec{\Leftarrow }}$) We define a canonical game form and show that it implements $\varphi$ in weak SDNE.

For each $i\in N,$ let $S_i \equiv \mathcal {P}^N\times \Delta (A)\times \Delta (A) \times \mathbb N_+.$ Denote a generic member of $S_i$ by $s_i\equiv (P^i,\pi ^i,\sigma ^i,n^i)$. Define $h:S\rightarrow \Delta (A)$ by setting, for each $s\in S$,

$$\begin{aligned} h(s) \equiv \left\{ \begin{array}{ll} \pi &{}\quad \text {if for each}\,i\in N,\, s_i = (P,\pi ,\sigma ,n)\,\text {and}\, \pi \in \varphi (P)\\ \sigma ^i&{}\quad \left\{ \begin{array}{l}\text {if there is }i\in N \text {such that} \\ \text {for each }j\in N {\setminus } \{i\}, s_j = (P,\pi ,\sigma ,n) \ne s_i\\ \quad =(P^i,\pi ^i,\sigma ^i,n^i) \\ \text {and } \sigma ^i\notin U^{sd}(P_i,\pi )\cup D,\end{array}\right. \\ \\ \pi &{} \quad \left\{ \begin{array}{l}\text {if there is }i\in N\text { such that} \\ \text {for each }j\in N {\setminus } \{i\}, s_j = (P,\pi ,\sigma ,n) \ne \\ \quad s_i=(P^i,\pi ^i,\sigma ^i,n^i) \\ \text {and } \sigma ^i\in U^{sd}(P_i,\pi )\cup D,\end{array}\right. \\ \\ \frac{1}{n^{i^*}+1}\sigma ^{i^*}+\frac{n^{i^*}}{n^{i^*}+1}\pi ^{i^*} &{} \quad \left\{ \begin{array}{l}\text {if there is a triple}\, i,j,k\in N \text {such that}\, s_i\ne s_j,\\ \quad s_i\ne s_k,s_j \ne s_k,\\ \text {and}\, \pi ^{i^*}\ne \sigma ^{i^*},\,\text {where}\, i^*\equiv \min \left\{ \text {arg}\max \limits _{j\in N} n^j\right\} ,\\ \quad \,\text {and }\end{array}\right. \\ \bar{\pi } &{}\quad \text {otherwise.} \end{array} \right. \end{aligned}$$

Let $\Gamma \equiv (S,h)$. Our goal is to show that for each $P\in \mathcal {P}^N$, $h(WNE^{sd}(\Gamma , P) = \varphi (P)$.

First, we show that $\varphi (P) \subseteq h(WNE^{sd}(\Gamma , P))$. Let $\pi \in \varphi (P)$. Let $s\in S$ be such that for each $i\in N$, $s_i = (P, \pi , \pi ,0 )$. For each $i\in N$ and each $s_i'\in S_i{\setminus } \{s_i\}$, either $h(s_i',s_{-i}) = h(s) = \pi$ or $h(s_i',s_{-i}) \notin U^{sd}(P_i,\pi )\cup D.$ Thus, $s\in WNE^{sd}(\Gamma , P)$. Consequently $\pi =h(s) \in h(WNE^{sd}(\Gamma , P))$.

Next, consider $\overset{\tiny \text {T}}{P}\in \mathcal {P}^N$ and $s\in WNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. We complete the proof by showing that $h(s)\in \varphi (\overset{\tiny \text {T}}{P})$. Let $\pi \equiv h(s)$.

Recall that for each $P_0\in \mathcal {P}$ and each $\pi \in \Delta (A)$, $U^{sd}(P_0,\pi )$ is closed and convex. Thus, for each $P_0'\in \mathcal {P}$ if $U^{sd}(P_0',\pi ) \subseteq U^{sd}(P_0,\pi )\cup D$ then $U^{sd}(P_0',\pi )\subseteq U^{sd}(P_0,\pi ).$

Case 1: For each $i\in N$, $s_i = (P,\pi ,\sigma , n)$ and $\pi \in \varphi (P)$. Then, for each $i\in N$, $U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )\subseteq U^{sd}(P_i,\pi )\cup D$: otherwise, there is $\sigma ^i\in \Delta (A)$ such that $\sigma ^i\ne \pi$ and $\sigma ^i \in U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ) {\setminus } (D\cup U^{sd}(P_i,\pi )).$ Let $s_i'\equiv (P,\pi ,\sigma ^i,n)$. Then $h(s_i',s_{-i}) = \sigma ^i$. So $h(s_i',s_{-i}) = \sigma ^i\in U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ){\setminus }\{\pi \} = U^{sd}(\overset{\tiny \text {T}}{P}_i,h(s)){\setminus }\{\pi \}$. This contradicts $s\in WNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. Since $\pi \in \varphi (P)$ and for each $i\in N$, $U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )\subseteq U^{sd}(P_i,\pi )$, by invariance to upper SD monotonic transformations, $\pi \in \varphi (\overset{\tiny \text {T}}{P})$.

Case 2: There is $i\in N$ such that, for each $j\in N{\setminus }\{i\}, s_j = (P,\pi ,\sigma , n)\ne s_i$. Let $\tilde{\pi }\equiv h(s)$. Then for each $j\in N{\setminus }\{i\}$, $\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{P}_j)}$: otherwise, let $\hat{s}_j\in S_j$ be such that ${\hat{n}^j} \equiv \max \{n^i,n\}+1$, ${\hat{\sigma }^j} \equiv \tilde{\pi }$, ${\hat{P}^j}\in \mathcal {P}^N$, and ${\hat{\pi }^j} = \delta ^{\tau (\overset{\tiny \text {T}}{P_j})}$. Then, $h(\hat{s}_j,s_{-j})$ $= \frac{1}{{\hat{n}^j}+1}\tilde{\pi }+ \frac{{\hat{n}^j}}{{\hat{n}^j}+1} \delta ^{\tau (\overset{\tiny \text {T}}{P_j})} \,{\overset{\tiny \text {T}}{P_j^{sd}}} \tilde{\pi }$. This contradicts $s\in WNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. Thus, for each $j\in N{\setminus }\{i\}$, $\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{P}_j)}\in D$.

Since $\tilde{\pi }\in ~D$, $\tilde{\pi }\ne \sigma ^i$ so, $\tilde{\pi }= \pi$. This implies that $U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi )\subseteq U^{sd}(P_i,\pi )\cup D$: Otherwise, there is $\sigma ^i\in \Delta (A)$ such that $\sigma ^i\ne \pi$ and $\sigma ^i\in U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ) {\setminus } (D\cup U^{sd}( P_i,\pi ))$. Let $s_i'\equiv (P,\pi ,\sigma ^i,n)$. Then, $h(s_i',s_{-i}) = \sigma ^i$. So $h(s_i',s_{-i}) = \sigma ^i \in U^{sd}(\overset{\tiny \text {T}}{P}_i,\pi ) = U^{sd}(\overset{\tiny \text {T}}{P}_i, h(s))$. This contradicts $s\in WNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$. Since $\pi \in \varphi (P)$ and for each $k\in N$, $U^{sd}(\overset{\tiny \text {T}}{P}_k,\pi )\subseteq U^{sd}(P_k,\pi )$, by invariance to upper SD monotonic transformations, $\pi \in \varphi (\overset{\tiny \text {T}}{P})$.

Case 3: There is a triple $i,j,k\in N$ such that $s_i \ne s_j, s_i\ne s_k,$ and $s_j\ne s_k$. Recall that $\pi \equiv h(s)$.

Case 3.1: $\pi = \frac{1}{n^{i^*}+1}\sigma ^{i^*}+\frac{n^{i^*}}{n^{i^*}+1}\pi ^{i^*}$    where $i^*\equiv \min \left\{ \text {arg}\max \limits _{j\in N} n^j\right\}$.

If $\pi ^{i^*}\,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \sigma ^{i^*}$, then let $\hat{s}_{i^*} \equiv (P^{i^*},\pi ^{i^*},\sigma ^{i^*},n^{i^*}+1)$.

If $\sigma ^{i^*}\,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \pi ^{i^*}$, then let $\hat{s}_{i^*} \equiv (P^{i^*},\sigma ^{i^*},\pi ^{i^*},n^{i^*}+1)$. Note that $\sigma ^{i^*}$ and $\pi ^{i^*}$ are interchanged.

Otherwise, let $\hat{s}_{i^*} \equiv (P^{i^*}, \hat{\pi }^{i^*}, \sigma ^{i^*},\hat{n}^{i^*})$, where $\hat{\pi }^{i^*} = \delta ^{\tau (\overset{\tiny \text {T}}{P}_{i^*})}$ and $\hat{n}^{i^*}$ is large enough that $\hat{n}^{i^*} > n^{i^*}$ and $\frac{1}{\hat{n}^{i^*}+1}\sigma ^{i^*}+\frac{\hat{n}^{i^*}}{\hat{n}^{i^*}+1}\hat{\pi }^{i^*} \,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \pi$.

For each of the above cases, $h(\hat{s}_{i^*}, s_{-i^*}) \,{\overset{\tiny \text {T}}{P^{sd}_{i^*}}} \pi = h(s)$. This contradicts $s\in WNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$.

Case 3.2: $\pi = \bar{\pi }$. Let $\hat{s}_{i^*} \equiv (\hat{P}^{i^*},\hat{\pi }^{i^*},\hat{\sigma }^{i^*},\hat{n}^{i^*}) \in S_{i^*}$ be such that $\hat{\pi }^{i^*} = \delta ^{\tau (P_{i^*})}$, $\hat{\sigma }^{i^*}(\ne \hat{\pi }^{i^*}) \,{\overset{\tiny \text {T}}{ P_{i^*}^{sd}}} \bar{\pi }$, and $\hat{n}^{i^*} \equiv {\max _{l\in N} n^l}+2$. Then, $h(\hat{s}_{i^*}, s_{-i^*}) = \frac{1}{\hat{n}^{i^*}}\hat{\sigma }^{i^*}+\frac{\hat{n}^{i^*}}{\hat{n}^{i^*}+1}\hat{\pi }^{i^*} \,{\overset{\tiny \text {T}}{P_{i^*}^{sd}}} \bar{\pi }= h(s)$. This contradicts $s\in WNE^{sd}(\Gamma , \overset{\tiny \text {T}}{P})$.

Implementation in expected utility NE

(${\varvec{\Rightarrow }}$) Suppose that $\varphi$ is implementable in expected utility NE. Suppose that it is implemented by $\Gamma \equiv (S,h)$. Let $u\in \mathcal {U}^N$ and $\pi \in \varphi \circ \rho (u)$. Then, there is $s\in NE(\Gamma , u)$ such that $h(s)=\pi$. Since $s\in NE(\Gamma , u)$, for each $i\in N$ and each $s_i'\in S_i$, $h(s_i',s_{-i})\in L^{eu}(u_i,\pi )$.

Let $u'\in \mathcal {U}^N$ be such that for each $i\in N$, $L^{eu}(u_i',\pi )\supseteq L^{eu}(u_i,\pi )$. Then, for each $s_i'\in S_i$, $h(s_i',s_{-i})\in L^{eu}(u_i',\pi )$. Thus, $s\in NE(\Gamma ,u)$ so $\pi =h(s) \in \varphi \circ \rho (u')$.

(${\varvec{\Leftarrow }}$) We define a canonical game form and show that it implements $\varphi$ in expected utility NE.

For each $i\in N,$ let $S_i \equiv \mathcal {U}^N\times \Delta (A)\times \Delta (A) \times \mathbb N_+.$ Denote a generic member of $S_i$ by $s_i\equiv (u^i,\pi ^i,\sigma ^i,n^i)$. Define $h:S\rightarrow \Delta (A)$ by setting, for each $s\in S$,

$$\begin{aligned} h(s) \equiv \left\{ \begin{array}{ll} \pi &{}\quad \text {if for each} i\in N,\, s_i = (P,\pi ,\sigma ,n)\, \text {and}\\ &{}\quad \pi \in \varphi \circ \rho (u)\\ \sigma ^i&{} \quad \left\{ \begin{array}{l}\text {if there is }i\in N\text { such that} \\ \text {for each }j\in N {\setminus } \{i\}, s_j = (u,\pi ,\sigma ,n) \ne s_i\\ \quad =(u^i,\pi ^i,\sigma ^i,n^i) \\ \text {and } \sigma ^i\in L^{eu}(u_i,\pi ){\setminus } D,\end{array}\right. \\ \\ \pi &{} \quad \left\{ \begin{array}{l}\text {if there is }i\in N\text { such that} \\ \text {for each }j\in N {\setminus } \{i\}, s_j = (u,\pi ,\sigma ,n) \ne s_i\\ \quad =(u^i,\pi ^i,\sigma ^i,n^i) \\ \text {and } \sigma ^i\notin L^{eu}(u_i,\pi ){\setminus } D,\end{array}\right. \\ \\ \frac{1}{n^{i^*}+1}\sigma ^{i^*}+\frac{n^{i^*}}{n^{i^*}+1}\pi ^{i^*} &{}\quad \left\{ \begin{array}{l}\text {if there is a triple} i,j,k\in N \text {such that}\, s_i\ne s_j,\\ \quad s_i\ne s_k,s_j \ne s_k, \\ \text {and} \pi ^{i^*}\ne \sigma ^{i^*}, \text {where} i^*\equiv \min \{\text {arg}\max \limits _{j\in N} n^j\}, \text {and }\end{array}\right. \\ \bar{\pi } &{} \quad \text {otherwise.} \end{array} \right. \end{aligned}$$

Let $\Gamma \equiv (S,h)$. Our goal is to show that for each for each $u\in \mathcal {U}^N$, $h(NE(\Gamma , u)) = \varphi \circ \rho (u)$.

First, we show that $\varphi \circ \rho (u) \subseteq h(NE(\Gamma , u))$. Let $\pi \in \varphi \circ \rho (u)$. Let $s\in S$ be such that for each $i\in N$, $s_i = (u, \pi , \pi ,0 )$. For each $i\in N$ and each $s_i'\in S_i{\setminus } \{s_i\}$, either $h(s_i',s_{-i}) = h(s) = \pi$ or $h(s_i',s_{-i}) \in L^{eu}(u_i,\pi ){\setminus } D.$ Thus, $s\in NE(\Gamma , u)$. Consequently $\pi =h(s) \in h(NE(\Gamma , u))$.

Next, consider $\overset{\tiny \text {T}}{u}\in \mathcal {U}^N$ and $s\in NE(\Gamma , \overset{\tiny \text {T}}{u})$. We complete the proof by showing that $h(s)\in \varphi \circ \rho (\overset{\tiny \text {T}}{u})$. Let $\pi \equiv h(s)$,

Recall that for each $u_0\in \mathcal {U}$ and each $\pi \in \Delta (A)$, $L^{eu}(u_0,\pi )$ is closed and convex. Thus, for each $u_0'\in \mathcal {U}$ if $L^{eu}(u_0',\pi ) \supseteq L^{eu}(u_0,\pi ){\setminus } D$ then $L^{eu}(u_0',\pi )\supseteq L^{eu}(u_0,\pi ).$

Case 1: For each $i\in N$, $s_i = (u,\pi ,\sigma , n)$ and $\pi \in \varphi \circ \rho (u)$. Then, for each $i\in N$, $L^{eu}(\overset{\tiny \text {T}}{u}_i,\pi )\supseteq L^{eu}(u_i,\pi ){\setminus } D$: otherwise, there is $\sigma ^i\in \Delta (A)$ such that $\sigma ^i \in L^{eu}(u_i,\pi ){\setminus } (D\cup L^{eu}(\overset{\tiny \text {T}}{u}_i,\pi )).$ Let $s_i'\equiv (u,\pi ,\sigma ^i,n)$. Then $h(s_i',s_{-i}) = \sigma ^i$. So $h(s_i',s_{-i}) = \sigma ^i$ where $\overset{\tiny \text {T}}{U}_i(\sigma ^i) > \overset{\tiny \text {T}}{U}_i(\pi )$. This contradicts $s\in NE(\Gamma , \overset{\tiny \text {T}}{u})$. Since $\pi \in \varphi \circ \rho (u)$ and for each $i\in N$, $L^{eu}(\overset{\tiny \text {T}}{u}_i,\pi )\supseteq L^{eu}(u_i,\pi )$, by invariance to Maskin monotonic transformations, $\pi \in \varphi \circ \rho (\overset{\tiny \text {T}}{u})$.

Case 2: There is $i\in N$ such that, for each $j\in N{\setminus }\{i\}, s_j = (u,\pi ,\sigma , n)\ne s_i$. Let $\tilde{\pi }\equiv h(s)$. Then for each $j\in N{\setminus }\{i\}$, $\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{u}_j)}$:²⁵ otherwise, let $\hat{s}_j\in S_j$ be such that ${\hat{n}^j} \equiv \max \{n^i,n\}+1$, ${\hat{\sigma }^j} \equiv \tilde{\pi }$, ${\hat{u}^j}\in \mathcal {U}^N$, and ${\hat{\pi }^j} = \delta ^{\tau (\overset{\tiny \text {T}}{u_j})}$. Then, since $h(\hat{s}_j,s_{-j}) = \frac{1}{{\hat{n}^j}+1}\tilde{\pi }+ \frac{{\hat{n}^j}}{{\hat{n}^j}+1} \delta ^{\tau (\overset{\tiny \text {T}}{u_j})}$, $\overset{\tiny \text {T}}{U}_j(h(\hat{s}_j,s_{-j})) > \overset{\tiny \text {T}}{U}_j(\tilde{\pi })$. This contradicts $s\in NE(\Gamma , \overset{\tiny \text {T}}{u})$. Thus, for each $j\in N{\setminus }\{i\}$, $\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{u}_j)}\in D$.

Since $\tilde{\pi }\in ~D$, $\tilde{\pi }\ne \sigma ^i$ so, $\tilde{\pi }= \pi$. This implies that $L^{eu}(\overset{\tiny \text {T}}{u}_i,\pi )\supseteq L^{eu}(u_i,\pi ){\setminus } D$: Otherwise, there is $\sigma ^i\in \Delta (A)$ such that $\sigma ^i\in L^{eu}( u_i,\pi ) {\setminus } (D\cup L^{eu}(\overset{\tiny \text {T}}{u}_i,\pi )))$. Let $s_i'\equiv (u,\pi ,\sigma ^i,n)$. Then, $h(s_i',s_{-i}) = \sigma ^i$ where $\overset{\tiny \text {T}}{U}_i(\sigma ^i)>\overset{\tiny \text {T}}{U}_i(\pi ).$ This contradicts $s\in NE(\Gamma , \overset{\tiny \text {T}}{u})$. Since $\pi \in \varphi \circ \rho (u)$ and for each $k\in N$, $L^{eu}(\overset{\tiny \text {T}}{u}_k,\pi )\supseteq L^{eu}(u_k,\pi )$, by invariance to Maskin monotonic transformations, $\pi \in \varphi \circ \rho (\overset{\tiny \text {T}}{u})$.

Case 3: There is a triple $i,j,k\in N$ such that $s_i \ne s_j, s_i\ne s_k,$ and $s_j\ne s_k$. Recall that $\pi \equiv h(s)$.

Case 3.1: $\pi = \frac{1}{n^{i^*}+1}\sigma ^{i^*}+\frac{n^{i^*}}{n^{i^*}+1}\pi ^{i^*}$    where $i^*\equiv \min \{\text {arg}\max \limits _{j\in N} n^j\}$.

If $\overset{\tiny \text {T}}{U}_{i^*} (\pi ^{i^*}) > \overset{\tiny \text {T}}{U}_{i^*}( \sigma ^{i^*})$, then let $\hat{s}_{i^*} \equiv (u^{i^*},\pi ^{i^*},\sigma ^{i^*},n^{i^*}+1)$.

If $\overset{\tiny \text {T}}{U}_{i^*} (\sigma ^{i^*}) > \overset{\tiny \text {T}}{U}_{i^*}(\pi ^{i^*})$, then let $\hat{s}_{i^*} \equiv (u^{i^*},\sigma ^{i^*},\pi ^{i^*},n^{i^*}+1)$. Note that $\sigma ^{i^*}$ and $\pi ^{i^*}$ are interchanged.

Otherwise, let $\hat{s}_{i^*} \equiv (u^{i^*}, \hat{\pi }^{i^*}, \sigma ^{i^*},\hat{n}^{i^*})$, where $\hat{\pi }^{i^*} = \delta ^{\tau (\overset{\tiny \text {T}}{u}_{i^*})}$ and $\hat{n}^{i^*}$ is large enough that $\hat{n}^{i^*} > n^{i^*}$ and $\overset{\tiny \text {T}}{U}_{i^*}\left( \frac{1}{\hat{n}^{i^*}+1}\sigma ^{i^*}+\frac{\hat{n}^{i^*}}{\hat{n}^{i^*}+1}\hat{\pi }^{i^*}\right) >\overset{\tiny \text {T}}{U}_{i^*}( \pi )$.

For each of the above cases, $\overset{\tiny \text {T}}{U}_{i^*}(h(\hat{s}_{i^*}, s_{-i^*})) >\overset{\tiny \text {T}}{U}_{i^*}(\pi ) = \overset{\tiny \text {T}}{U}_{i^*}(h(s))$. This contradicts $s\in NE(\Gamma , \overset{\tiny \text {T}}{u})$.

Case 3.2: $\pi = \bar{\pi }$. Let $\hat{s}_{i^*} \equiv (\hat{u}^{i^*},\hat{\pi }^{i^*},\hat{\sigma }^{i^*},\hat{n}^{i^*}) \in S_{i^*}$ be such that $\hat{\pi }^{i^*} = \delta ^{\tau (u_{i^*})}$, $\hat{\sigma }^{i^*}(\ne \hat{\pi }^{i^*}) \notin L^{eu}(\overset{\tiny \text {T}}{u}_{i^*}, \bar{\pi })$, and $\hat{n}^{i^*} \equiv {\max _{l\in N} n^l}+2$. Then, since $h(\hat{s}_{i^*}, s_{-i^*}) = \frac{1}{\hat{n}^{i^*}}\hat{\sigma }^{i^*}+\frac{\hat{n}^{i^*}}{\hat{n}^{i^*}+1}\hat{\pi }^{i^*}$, $\overset{\tiny \text {T}}{U}_{i^*}(h(\hat{s}_{i^*}, s_{-i^*})) > \overset{\tiny \text {T}}{U}_{i^*}(\bar{\pi }) = \overset{\tiny \text {T}}{U}_{i^*}(h(s))$. This contradicts $s\in NE(\Gamma , \overset{\tiny \text {T}}{u})$.

C Proof of Proposition 4

(${\varvec{\bigcup }} {\varvec{\Phi }}$) Let $P\in \mathcal {P}^N$ and $\pi \in (\bigcup \Phi )(P)$. Then, there is $\phi \in \Phi$ such that $\pi \in \phi (P)$. Let $P'\in \mathcal {P}^N$ be a support monotonic transformation of P at $\pi$. By invariance to support monotonic transformations, $\pi \in \phi (P')$. Thus, $\pi \in (\bigcup \Phi )(P')$.

(${\varvec{\bigcap }}{\varvec{\Phi }}$) Let $P\in \mathcal {P}^N$ and $\pi \in (\bigcap \Phi )(P)$. Then, for each $\phi \in \Phi$, $\pi \in \phi (P)$. Let $P'\in \mathcal {P}^N$ be a support monotonic transformation of P at $\pi$. By invariance to support monotonic transformations, for each $\phi \in \Phi$, $\pi \in \phi (P')$. Thus, $\pi \in (\bigcap \Phi )(P')$.

(Convex combination) Let $P\in \mathcal {P}^N$, $\alpha \in (0,1)$, $\underline{\pi }\in \phi (P)$, and $\overline{\pi }\in \varphi (P)$. Let $\pi \equiv (\alpha \underline{\pi }+ (1-\alpha )\overline{\pi })\in (\alpha \phi +(1-\alpha )\varphi )(P)$. Let $P'\in \mathcal {P}^N$ be a support monotonic transformation of P at $\pi$. For each $i\in N$ and each $a\in A$, $\pi _i(a)=0$ if and only if $\underline{\pi }_{i}(a)=\overline{\pi }_{i}(a)=0$. Thus, $P'$ is a support monotonic transformation of P at both $\underline{\pi }$ and $\overline{\pi }$. By invariance to support monotonic transformations, $\underline{\pi }\in \phi (P')$ and $\overline{\pi }\in \varphi (P')$. Thus, $\pi \in (\alpha \phi +(1-\alpha )\varphi )(P')$.

D Implementability of other solutions

We list a few other solutions discussed in the literature. The following is less demanding than $F^{sd}$ as it drops the requirement of SD comparability.

Weak SD no-envy solution, ${{\varvec{WF}}^{{\varvec{sd}}}}$: For each $P\in \mathcal {P}^N$, $WF^{sd}(P)=\{\pi \in \Delta (F):$ for each $i\in N$, there is no $j\in N$ such that $\pi _j\,{P_i^{sd}}\pi _i\}$.

Let $\theta :N\rightarrow \{1,\ldots ,n\}$ be a one-to-one mapping from N to itself. Let $\Theta$ be the set of all such mappings. We define a class of single-valued solutions. Each member in this class is indexed by such an ordering and sequentially assigns each agent his most preferred object, among those remaining, with certainty.

Sequential priority solution associated with ${{\varvec{\theta }}{\varvec{\in }} {\varvec{\Theta }}}$, ${{\varvec{SP}}^{\varvec{\theta }}}$: For each $P\in \mathcal {P}^N$, $SP^\theta (P)\equiv \pi \in \Delta (F)$ defined as follows: $\pi _{\theta (1)}(\tau (P_{\theta (1)},O)) \equiv 1,$ $\pi _{\theta (2)}(\tau (P_{\theta (2)},O{\setminus }\tau (P_{\theta (1)}, O))) \equiv 1$, and so on.

The next solution is defined by uniformly randomizing over all n! possible elements of $\Theta$ and then applying the relevant sequential priority solution.

Random priority solution, RP: For each $P\in \mathcal {P}^N$, $RP(P)\equiv \frac{1}{n!}\sum _{\theta \in \Theta }SP^\theta (P)$.

Proposition 6

The solutions $WF^{sd}$, for each $\theta \in \Theta$, $SP^\theta$, and RP are implementable in Nash equilibria.

Proof

We prove that each of these solutions is invariant to support monotonic transformation.

${{WF}}^{{{sd}}}$: Let $P\in \mathcal {P}^N$ and $\pi \in WF^{sd}(P)$. Then, there is no pair $i,j\in N$ such that $\pi _j\,{P_i^{sd}} \pi _i$. That is, for each pair ${i,j\in N}$, either (1) $\pi _i=\pi _j$ or (2) there is $o\in O$ such that ${\pi _i(o) > 0}$ and $\pi _{i}(L(P_i,o))< \pi _{j}(L(P_i,o))$. Let $P'\in \mathcal {P}^N$ be a support monotonic transformation of P at $\pi$. Then, for each $k\in N$ and each $t\in O$ such that ${\pi _k(t)>0}$, $L(P_k,t)\subseteq L(P'_k,t)$. In case (2), $\pi _{j}(L(P'_i,o)) \ge \pi _{j}(L(P_i,o)) > \pi _{i}(L(P_i,o)) = \pi _{i}(L(P'_i,o)).$ The last equality comes from the fact that $P'$ is a support monotonic transformation of p at $\pi$ Thus, for each pair ${i,j\in N}$, either (1) $\pi _i=\pi _j$ or (2) there is $o\in O$ such that ${\pi _i(o) > 0}$ and $\pi _{i}(L(P'_i,o))< \pi _{j}(L(P_i',o))$, and so $\pi \in WF^{sd}(P')$. l ${({{SP}}^{{\theta }})_{{{\theta }}{{\in }} {{\Theta }}}}$ and RP: it is straightforward so we omit the proof. $\square$