# 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.

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1. 1.

By this we mean that probability of each of the following events is at least as likely (and at least one is more likely) under the first lottery as it is under the second: the top alternative is realized, one of the top two alternatives is realized, one of the top three alternatives is realized, and so on.

2. 2.

This is often referred to as the “probabilistic serial rule.”

3. 3.

We have adopted the terminology of Thomson (2013).

4. 4.

Note that in these two papers, a game form used to implement such solutions allows randomization over sure alternatives and strong SDNE is adopted as the equilibrium concept. However, as long as the game form implements the solution selecting sure alternatives, equilibrium outcomes are sure alternatives. That is, lotteries only appear off the equilibrium path.

5. 5.

This is equivalent to stating that for each $$a\in A$$, $$\pi '(L(P_0,a)) \ge \pi (L(P_0,a))$$ with at least one strict inequality and equality for the best alternative in A according to $$P_0$$.

6. 6.

One way to complete this relation is to treat non-comparability as indifference. The resulting complete relation, however, will not be transitive. For example, consider $$P_0$$ such that $$a\,P_0 b\,P_0 c$$ and three lotteries

$$\pi _0=(0.4, 0, 0.6)$$, $$\pi _0'=(0,1,0)$$, and $$\pi _0''=(0.6,0,0.4)$$. Neither $$\pi _0$$ and $$\pi _0'$$ nor $$\pi _0'$$ and $$\pi _0''$$ are comparable. However, $$\pi _0''~P_0^{sd}~\pi _0$$.

7. 7.

That is, for each $$u_0\in \mathcal {U}$$, $$u_0\in \mathcal U(\rho (u_0))$$.

8. 8.

We adopt the terminology of Thomson (2013). By “Maskin-invariance” we mean the property known in the literature as “Maskin-monotonicity.” Thus, we refer to the new conditions that we introduce as “invariance” conditions rather than as “monotonicity” conditions.

9. 9.

For each $$\pi \in \Delta (A)$$, supp$$(\pi )=\{a\in A: \pi (a)>0\}$$.

10. 10.

That there are no monotonic transformations of a preference relation on the interior of the simplex renders Maskin-invariance vacuous on all but the boundary. This is a key insight for the “anything goes” results on virtual implementationMatsushima 1988; Abreu and Sen 1991.

11. 11.

Note that $$X_i$$ is the set of deterministic allocations that i may receive.

12. 12.

To see that the environment of Sect. 2 is a special case, note that we can set, for each $$i\in N, X_i \equiv A$$ and let $$F\equiv \{x\in A^N: \text { for each pair }i,j\in N, x_i = x_j\}.$$

13. 13.

Regularity is a mild requirement. It is vacuous if every agent can be assigned only one object and there are exactly as many objects as there are agents. In general, it is implied by very weak efficiency requirements: Suppose that every agent but i receives some objects with probability one. Since the remaining objects are available with certainty, what i receives should be certain as well.

14. 14.

This is often called ordinal efficiency. We have adopted the terminology and the notation of Thomson (2013).

15. 15.

If for each $$i\in N$$ and each $$a\in A$$, $$\omega _{i}(a)=\frac{1}{n}$$, this is the “SD equal division lower bound” solution (Heo 2014).

16. 16.

See Bogomolnaia and Moulin (2001) for a formal definition of this solution.

17. 17.

Proposition 5 (2) does not generalize to more than three agents.

18. 18.

See the proof of Proposition 2 in Bogomolnaia and Moulin (2001).

19. 19.

We also check implementability of some other solutions in “Appendix D”.

20. 20.

When agents may receive more than one object, even stronger impossibility results hold (Kojima 2009; Kasajima 2011; Heo 2014).

21. 21.

Equivalently: $$|\text {supp}(\sigma ^i)| \ne 1 \text { and } \pi \,{R_i^{sd}} \sigma ^i.$$

22. 22.

This is a variant of the canonical game form by Maskin (1999).

23. 23.

Otherwise, there is $$a\in A$$ such that $$\delta ^a\in L^{sd}(P_0,\pi )$$ but $$\delta ^a\notin L^{sd}(P_0',\pi )$$. Therefore, $$\delta ^a\ne \pi$$. Since $$L^{sd}(P_0,\pi )$$ is closed and convex and $$\pi \ne \delta ^a$$, for each $$\alpha \in (0,1)$$, $$\alpha \delta ^a+(1-\alpha )\pi \in L^{sd}(P_0,\pi ){\setminus } D \subseteq L^{sd}(P_0',\pi )$$, where the last inclusion relation is from the assumption. Since $$\delta ^a\notin L^{sd}(P'_0,\pi )$$ and $$L^{sd}(P'_0,\pi )$$ is closed, there is $$\beta \in (0,1)$$ such that $$\beta \delta ^a+(1-\beta )\pi \in L^{sd}(P_0,\pi ){\setminus }(L^{sd}(P'_0,\pi ){\setminus } D)$$, a contradiction.

24. 24.

That is, $$\tilde{\pi }$$ is the top alternative for each $$j\in N{\setminus }\{i\}$$ under $$\overset{\tiny \text {T}}{P}_j$$.

25. 25.

For $$u_0\in \mathcal {U}$$, we write $$\tau (u_0)$$ to mean $$\tau (\rho (u_0))$$.

## References

1. Abreu D, Sen A (1991) Virtual Implementation in nash equilibrium. Econometrica 59:997–1021

2. Athanassoglou S, Sethuraman J (2011) House allocation with fractional endowments. Int J Game Theory 40:481–513

3. Benoît J-P, Ok EA (2008) Nash implementation without no-veto power. Games Econ Behav 64:51–67

4. Bochet O (2007) Nash implementation with lottery mechanisms. Soc Choice Welf 28:111–125

5. Olivier B, Sakai T (2005) Nash implementation in stochastic social choice. Working paper, Yokohama National University

6. Bogomolnaia A, Moulin H (2001) A new solution to the random assignment problem. J Econ Theory 88:233–260

7. Budish E (2011) The combinatorial assignment problem: approximate competitive equilibrium from equal incomes. J Polit Econ 119(6):1061–1103

8. Budish E, Che Y-K, Kojima F, Milgrom P (2013) Designing random allocation mechanisms: theory and applications. Am Econ Rev 103(2):585–623

9. Dasgupta PS, Hammond PJ, Maskin ES (1979) The implementation of social choice rules: some general results on incentive compatibility. Rev Econ Stud 46:185–216

10. d’Aspremont C, Gérard-Varet L-A (1979) Incentives and incomplete information. J Publ Econ 11(1):25–45

11. Ehlers L, Masso J (2007) Incomplete information and singleton cores in matching markets. J Econ Theory 136(1):587–600

12. Özgün E, Kesten O (2010) On the ordinal nash equilibria of the probabilistic serial mechanism. Working paper, Carnegie Mellon University

13. Foley D (1967) Resource allocation and the public sector. Yale Econ Essays 7:45–98

14. Gibbard A (1977) Manipulation of schemes that mix voting with chance. Econometrica 45(3):665–681

15. Harris M, Townsend RM (1981) Resource allocation under asymmetric information. Econometrica 49(1):33–64

16. Heo EJ (2014) Probabilistic assignment problem with multi-unit demands: a generalization of the serial rule and its characterization. J Math Econ 54:40–47

17. Hylland A, Zeckhauser R (1979) The efficient allocation of individuals to positions. J Polit Econ 87(2):293–314

18. Kasajima Y (2011) More on the probabilistic assignment of indivisible goods when agents receive several. Working paper, Hiroshima Shudo University

19. Kojima F (2009) Random assignment of multiple indivisible objects. Math Soc Sci 57:134–142

20. Majumdar D, Sen A (2004) Ordinally bayesian incentive compatible voting rules. Econometrica 72:523–540 03

21. Maskin E (1999) Nash equilibrium and welfare optimality. Rev Econ Stud 66:23–38

22. Matsushima H (1988) A new approach to the implementation problem. J Econ Theory 45:128–144

23. Myerson RB (1979) Incentive compatibility and the bargaining problem. Econometrica 47(1):61–73

24. Pais J (2008) Incentives in decentralized random matching markets. Games Econ Behav 64:632–649

25. Peivandi A (2014) Random allocation of bundles. Working paper, Northwestern University

26. Sen A (1995) The implementation of social choice functions via social choice correspondences: a general formulation and a limit result. Soc Choice Welf 12:277–292

27. Thomson W (1999) Monotonic extensions on economic domains. Rev Econ Des 4:13–33

28. Thomson W (2013) Strategy-proof allocation rules. Unpublished monograph, University of Rochester

## Author information

Authors

### Corresponding author

Correspondence to Eun Jeong Heo.

We thank William Thomson for his guidance, encouragement, and support. We are grateful to the editor and the two anonymous referees for their constructive comments and suggestions. We also thank Wonki Cho, Battal Doǧan, Paula Jaramillo, and John Weymark for helpful comments and discussions.

## 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(aP) 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}

Let $$\Gamma \equiv (S,h)$$.Footnote 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 $$L^{sd}(P_0',\pi )\supseteq L^{sd}(P_0,\pi ).$$ Footnote 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 $$j\in N{\setminus }\{i\}$$, $$\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{P}_j)}$$.Footnote 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 $$\pi \equiv h(s)$$.

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 “$$\sigma ^i\in L^{eu}(u_i,\pi ){\setminus } D$$.”

### 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 $$j\in N{\setminus }\{i\}$$, $$\tilde{\pi }= \delta ^{\tau (\overset{\tiny \text {T}}{u}_j)}$$:Footnote 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. $$\square$$

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