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Pure Nash Equilibria under the Affine Mechanism: A Potential Game of Exaggeration
Jason Jisen Li, Young Wu, Yancheng Zhu, Jin-yi Cai, Xiaojin Zhu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 7/9/2026, 9:03:11 AM
Summary
This paper analyzes the Affine Mechanism Game (AMG), a generalization of the mean mechanism used for numerical aggregation. It proves that AMG is a potential game where pure Nash equilibria always exist. The analysis demonstrates that in any pure Nash equilibrium, all players except at most one will strategically misreport their values to the maximum extent possible, highlighting the inevitability of extreme exaggeration. The study further characterizes these equilibria in both complete-information and Bayesian settings, and shows that high-dimensional action spaces can be decomposed into independent one-dimensional problems with closed-form solutions.
Entities (8)
Relation Signals (6)
University of Wisconsin-Madison → affiliatedwith → Jason Jisen Li
confidence 98% · 11institutetext: University of Wisconsin - Madison ... Jason Jisen Li
Affine Mechanism Game → isa → Potential Game
confidence 97% · We show GG is a potential game and study its pure strategy Nash equilibria.
Mean Mechanism → isaspecialcaseof → Affine Mechanism Game
confidence 96% · the affine mechanism, of which the mean is a special case.
Pure Nash Equilibrium → involves → Extreme Exaggeration
confidence 95% · In each pure NE, all but at most one player must exaggerate their misreporting to the maximum extent possible.
Affine Mechanism Game → generalizesto → Bayesian Game
confidence 94% · When players do not know other players’ types, we provide a similar Bayesian NE characterization.
Affine Mechanism Game → uses → Squared 2-norm Loss
confidence 93% · We consider the squared 2-norm loss: ℓi(𝐱):=‖𝐱^−𝐭i‖22
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Abstract
Abstract:The mean mechanism is known to be non-incentive-compatible, namely, rational players are incentivized to misreport their values. Despite this game-theoretic issue, the mean mechanism is prevalent in practice due to its other desirable properties. We give a full characterization of pure Nash equilibria--how the players will misreport--for the affine mechanism, of which the mean is a special case. Furthermore, we characterize both complete-information and Bayesian games under the affine mechanism. Our results highlight the inevitability of extreme exaggeration in such games.
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- Source: https://arxiv.org/abs/2606.29010v1
- Canonical: https://arxiv.org/abs/2606.29010v1
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11institutetext: University of Wisconsin - Madison Pure Nash Equilibria under the Affine Mechanism: A Potential Game of Exaggeration Jason Jisen Li Young Wu Yancheng Zhu Jin-yi Cai Xiaojin Zhu Abstract The mean mechanism is known to be non-incentive-compatible, namely, rational players are incentivized to misreport their values. Despite this game-theoretic issue, the mean mechanism is prevalent in practice due to its other desirable properties. We give a full characterization of pure Nash equilibria–how the players will misreport–for the affine mechanism, of which the mean is a special case. Furthermore, we characterize both complete-information and Bayesian games under the affine mechanism. Our results highlight the inevitability of extreme exaggeration in such games. 1 Introduction Consider the classic numerical aggregation problem: n players each report a number x1…xnx_1… x_n. A mechanism aggregates these into a single number x x, e.g., using the mean or the median function. Player i’s utility depends on how close xix_i and x x are (assuming the reported xix_i is truthful). There has been a sharp divide in theory and practice: • In theory, it is well understood that on single-peaked preference domains, the strategyproof rules are exactly the generalized median (phantom-voter) aggregation mechanisms ([Moulin, 1980]). In contrast, the mean mechanism is the textbook example of non-incentive-compatibility: players will have an incentive to misreport their numbers (Procaccia and Tennenholtz [2013], Chan et al. [2021]). • In practice, the mean remains the dominant aggregation rule when participants submit numerical reports. Corporate and academic performance evaluations such as paper reviews or student evaluations of teaching are almost universally aggregated with the mean. In congressional committee legislative voting over continuous variables (e.g., deciding the specific dollar amount for a corporate tax rate), final compromises are routinely arrived at via split-the-difference arithmetic averages (Martin [2001]). Many expert judgment procedures, such as engineering risk assessment or technology forecasting, aggregate experts by equal-weight averaging (Cooke [1991]). Using the mean is often not out of ignorance: it is a conscious choice to trade strategyproofness for better estimates under the L2 utilitarian social welfare function (Gershkov et al. [2017]), for fairness when the policymaker believes extreme outliers deserve a voice even if it invites cheating (Maskin [2008]), and for interpretability, where explaining a median mechanism with phantom agents to the public is politically unpalatable (Walsh [2025]). When the stake is high, often practitioners patch up the mean mechanism via trimmed mean (as in skating judging) or weighted average, rather than completely abandoning the mean mechanism for the median. Therefore, despite being non-strategyproof, the mean mechanism is practiced ubiquitously. While it was well-known that strategic players will misreport under the mean mechanism, it was less clear how they misreport. A better understanding is important for detecting strategic behaviors, and for making an informed decision when choosing a mechanism to balance welfare efficiency and control cheating. When players’ strategic misreporting is viewed as an attempt to “game the system”, it also has security implications. This paper gives a full characterization of how the players rationally misreport under affine mechanisms, including the mean mechanism. Our main contributions are: 1. We prove that pure Nash Equilibria (pure NEs) always exist under the mean mechanism. This is done via identifying its potential function. 2. In each pure NE, all but at most one player must exaggerate their misreporting to the maximum extent possible. 3. In decomposable high-dimensional action spaces, the pure NEs are also decomposable into individual 1D problems and admit closed-form solutions. 4. When players do not know other players’ types, we provide a similar Bayesian NE characterization. 2 Related Work Our work is closely related to the facility location problem. Moulin [1980] and Procaccia and Tennenholtz [2013] introduce the mechanism design problem for facility location, where the mechanism designer selects a facility location as a function of the players’ reported preferences in a way that rational players will truthfully report their preferences. In particular, they note that direct averaging of players’ reports will incentivize them to misreport. Berga [2002] extends the problem to a multidimensional domain similar to our setting, but it focuses on an incentive-compatible mechanism called the generalized median voter scheme. Our paper is not to advocate for the mean mechanism. But given its practical relevance, we focus on characterizing its pure NEs. Schummer and Vohra [2002], Filimonov and Meir [2023] study the facility location problem on graphs and trees; Goel and Hann-Caruthers [2020], Zampetakis and Zhang [2023], Barak et al. [2024] study the approximation mechanisms in multidimensional Euclidean spaces, which use similar techniques and are related to our reduction of the multidimensional problem into multiple single-dimensional ones, but they also focus on decomposition of the generalized median type mechanisms instead of the mean mechanism. An additional justification for the mean mechanism stems from a decision-theoretic perspective: the optimality of the Bayes estimator for the posterior mean. Under quadratic loss and Gaussian distributional assumptions, weighted averaging is not only commonly used but also optimal. This is empirically studied and commonly assumed in the social science literature, including models by DeGroot and Mortera [1991], Romeijn [2024], which argue that averaging, or what they call linear opinion pooling, is a form of optimal Bayesian updating based on other people’s opinions, assuming truthfulness. They interpret the weights in the averaging process as trust coefficients, which our model also uses. In the economics and forecasting literature, simple averaging often performs better than more complex methods across various economic forecasting tasks, as summarized in Clemen [1989], Wang et al. [2023]. One interesting line of research observed human exaggeration behaviors in the context of political polarization and media bias, including Jost et al. [2022], Kubin and Von Sikorski [2021], Van Bavel et al. [2021]. Although these studies do not directly link the behavior to a simple mean mechanism, they provide empirical evidence of exaggeration as an equilibrium behavior, and the non-incentive-compatibility of humans’ opinion aggregation mechanisms could be an important contributing factor. 3 Problem Setup We consider the affine mechanism in an n-player simultaneous-move game. The players have a common continuous action space ⊂ℝdX ^d which is compact and convex. Let i∈x_i be the action of the iith player for i∈[n]≔1,2,…,ni∈[n] \1,2,...,n\. A joint action, or pure strategy profile, =(1,…,n)x=(x_1,…,x_n) denotes the simultaneous action choice of all players. As in standard literature, we also write =(i,−i)x=(x_i,x_-i) when we want to emphasize player i. The mean mechanism aggregates to various degrees the inputs it receives from all players. In this paper, we consider a slight generalization of the mean, namely the affine mechanism of the form ^:=w00+∑i=1nwii, x:=w_0x_0+ _i=1^nw_ix_i, (1) where wi>0,i∈[n]w_i>0,i∈[n] is a real-valued (not necessarily normalized) weight that signifies how much influence player i has on the mechanism. 0∈x_0 is a bias term that, together with w0∈ℝw_0 , denotes a fixed, constant “background” influence that is beyond the control of the n players. The affine mechanism (1) is common knowledge to all players. The n players each have a loss (negative utility) function ℓi _i defined by a target i∈ℝdt_i ^d. Note the targets are not restricted to X. The goal of player i is to drive the mechanism’s x close to it_i. We consider the squared 2-norm loss: ℓi():=‖^−i‖22=‖w00+∑j=1nwjj−i‖22. _i(x):=\| x-t_i\|_2^2=\|w_0x_0+ _j=1^nw_jx_j-t_i\|_2^2. (2) As rational agents, the players want to selfishly minimize their own ℓi() _i(x). The players compete with each other when their targets 1,…,nt_1,…,t_n are distinct (though our setup allows some or all targets to overlap, too). The fact that the mechanism’s x is defined by the joint action x couples the players together in a general-sum game. The above narrative can be abstracted into the following formal affine mechanism game, Definition 1(Affine Mechanism Game (AMG)) The Affine Mechanism Game is an n-player general-sum game G=(n,,ℓii=1n)G= (n,X,\ _i\_i=1^n ), where n is the number of players, ⊂ℝdX ^d is a compact and convex action space, and ℓi:n↦ℝ _i:X^n taking the form of equation (2) is player i’s loss function. In the rest of this section, we assume that the targets (and hence loss functions) are common knowledge to all players. We show G is a potential game and study its pure strategy Nash equilibria. We then characterize the properties of these pure NEs. In section 4 we relax this assumption and study the resulting Bayesian game. 3.1 AMG is a Potential Game Definition 2 A pure strategy Nash equilibrium of the game G=(n,,ℓii=1n)G= (n,X, \ _i \_i=1^n ) is a strategy profile ∈nx ^n satisfying ℓi(i,−i)≤ℓi(,−i),∀∈,i∈[n] _i (x_i,x_-i )≤ _i (y,x_-i ),∀\;y ,i∈[n] . Our AMG satisfies the Debreu-Glicksberg-Fan theorem and pure NEs exist. To find the pure NEs explicitly, we turn to potential games. Potential games are games with a special structure that allow strong results on pure Nash equilibria (Monderer and Shapley [1996]). We now show AMG is a potential game: Theorem 3(Potential Game) The AMG G is a potential game with the potential function ϕ(1,…,n):=‖∑i=0nwii‖22−2∑i=1nwii⊤i.φ(x_1,…,x_n):= \| _i=0^nw_ix_i \|_2^2-2 _i=1^nw_it_i x_i. (3) As a potential game, AMG has pure NEs. Section 3.2 will be dedicated to the structure of these pure NEs. First, we remark on algorithms for finding pure NEs of AMG. Due to Proposition 5, any convex optimization algorithm that minimizes the convex function ϕφ over the convex set nX^n with strong guarantees can be utilized to find a pure NE (Boyd and Vandenberghe [2004]). Meanwhile, in the game theory community, the best-response dynamics is a traditional algorithm for finding a pure NE in potential games (Roughgarden [2010]): Remark 4(Best-Response Dynamics) Starting from an arbitrary (0)∈nx (0 ) ^n, the best response sequence (t)t=1∞ \x (t ) \_t=1^∞, under standard coordinate descent conditions (Tseng [2001]), every limit point of the sequence is a minimizer of ϕφ, hence a pure Nash equilibrium of G, where i(t)=argminiℓi(i,−i(t−1))x (t )_i= argmin_x_i _i (x_i,x (t-1 )_-i ) if i=((t−1)modn)+1i=((t-1) n)+1, and i(t−1)x (t-1 )_i otherwise. One interesting observation that is relevant for AMG is that, under best response dynamics, no player needs to know other players’ targets. Concretely, the players may carry out the best response dynamics as a learning dynamics in a distributed fashion over time, with no two players simultaneously updating their actions. When player i updates its own action ix_i, it observes other player’s most recent actions −ix_-i but does not need to know their targets −it_-i. Therefore, the best response dynamics may offer a computational account of how players in the real world iteratively adjust their actions based on actions of other players without knowing the other players’ true intentions, and still reach an equilibrium. Note that minimizing the potential function ϕφ along the ix_i direction is equivalent to minimizing its own loss function ℓi _i: argminiϕ(i,−i)=argminiℓi(i,−i) argmin_x_iφ(x_i,x_-i)= argmin_x_i _i(x_i,x_-i). As a result, best response dynamics also correspond to cyclic coordinate descent on ϕφ ( Wright and Recht [2022]). 3.2 Structure of Pure Nash Equilibria in AMG We next show that ∈nx ^n is a pure NE of G if and only if x is a minimum of ϕφ restricted to the domain nX^n. Proposition 5(Pure NEs ⇔ minima) The set of pure Nash equilibria in G is pureNE(G)=argmin∈nϕ()pureNE(G)= argmin_x ^nφ(x), and moreover, the set of correlated equilibria is the set of mixtures of pure Nash equilibria, that is, CE(G)=ΔpureNE(G)CE(G)= (G). We remark that by definition X is compact and convex, thus nX^n is bounded and closed. The potential function ϕ()φ(x) may not have a global minimum on the extended domain ℝndR^nd (it could diverge to −∞-∞ there), but on nX^n it will have at least one minimum (perhaps on the boundary). In fact, we have the following guarantee. Corollary 6(Cardinality of pureNE(G)pureNE(G)) pureNE(G)pureNE(G) is non-empty and convex, that is, G has either one pure NE or infinite pure NEs. In particular, the second part of Proposition 5 and this Corollary imply that in the case of a unique pure strategy Nash equilibrium x (for example, in the 1D two-player case with distinct targets, which we expand on in the next subsection), then x is the unique Nash equilibrium and the unique correlated equilibrium. We provide a few illustrative examples of AMG in subsection 3.3. In AMG the pure NEs often seem to involve all players misrepresenting their true target to the mechanism (i.e. i≠ix_i _i). Furthermore, such misrepresentation often seems to take the form of extreme exaggeration, in the sense that a player’s rational action ix_i at any pure NE is often pushed to the boundary of the action space X so they cannot exaggerate the action further. These observations are almost true, but need a subtle correction: Given that the players’ targets 1…nt_1…t_n are all distinct, at any pure NE, extreme exaggeration is necessary for all (except perhaps one) players. In other words, there are two possibilities: Either all players are at the boundary of X, or one of them is in the interior of X. In the latter case, that special player (say player i∗i^*) will in general still need to exaggerate its target i∗≠i∗x_i^* _i^* to the mechanism; it is just that i∗x_i^* is in the interior of X and not at the boundary. Furthermore, player i∗i^* is the lucky winner in that the mechanism will end up at its target i∗t_i^*. Our next theorem precisely characterizes this phenomenon. Theorem 7(All-But-At-Most-One Extreme Exaggeration) If ii=1n \t_i \_i=1^n are all distinct, then every pure NE satisfies the property that |i∈[n]:i∈int|≤1 | \i∈[n]:x_i \;X \ |≤ 1. Conversely, in a NE, if i,j∈intx_i,x_j \;X, then i=jt_i=t_j. Furthermore, if i⋆∈intx_i \;X for some i⋆∈[n]i ∈ [n ], then ^=i⋆ x=t_i . 3.3 Computing Pure Nash Equilibria We now show how to find pure NEs in high dimensional action space by decomposing the problem. The base case is when X is in a one-dimensional space, under the assumption that it is compact and convex, we can write =[L,U]X= [L,U ], and in this case, we can fully characterize the set of all Nash equilibria as follows. Lemma 8(Nash Equilibria in 1D1D) When =[L,U]X= [L,U ], then there exists some t⋆∈ℝt , such that in every Nash equilibrium (x1,x2,…,xn) (x_1,x_2,...,x_n ) of AMG, we have x:xi=L if ti<t⋆xi:∑i:ti=t⋆wixi=t⋆−w0x0−∑i:ti<t⋆wiL−∑i:ti>t⋆wiU if ti=t⋆xi=U if ti>t⋆. x:. (4) In particular, when targets are sorted t1≤t2≤…≤tnt_1≤ t_2≤...≤ t_n, (LABEL:eq:odne) is equivalent to the existence of i¯ i and i¯ i such that in every NE x1=x2=…=xi¯=Lx_1=x_2=...=x_ i=L, xi¯=xi¯+1=…=xn=Ux_ i=x_ i+1=...=x_n=U, and if there are players with the target ti=t⋆t_i=t with i¯<i<i¯ i<i< i, then it is NE as long as their xix_i results in x^=t⋆ x=t . Remark 9 A simple algorithm to find an NE is to iterate over the following set, t⋆∈t1,t2,…,tn∪12(t1+t2),12(t2+t3),…,12(tn−1+tn)∪L,Ut ∈ \t_1,t_2,...,t_n \∪ \ 12 (t_1+t_2 ), 12 (t_2+t_3 ),..., 12 (t_n-1+t_n ) \∪ \L,U \, and for each candidate t⋆t , compare the value of the potential function ϕ(x)φ (x ) with x satisfying (LABEL:eq:odne), and the x that minimizes ϕφ is a NE. In general, it is difficult to find a closed form solution in higher dimensions, but for special X that is a Cartesian product of intervals (i.e. a hyper-rectangle), that is =∏k=1dkX= _k=1^dX_k with k=[Lk,Uk]X_k= [L_k,U_k ], we provide the following decomposition result, which implies an algorithm to solve d one-dimensional problems using Lemma 8 to obtain the NE coordinate-wise. Since the loss function is coordinate-wise-separable, ℓi()=∑k=1dℓik(k) _i(x)= _k=1^d _ik(x_k), where ℓik(k):=(^k−ik)2 _ik(x_k):=( x_k-t_ik)^2. We can define for each k∈[d]k∈[d], GkG_k with the action space k=[Lk,Uk]X_k=[L_k,U_k], loss ℓik _ik, and mechanism ^k=w00k+∑i=1nwiik x_k=w_0x_0k+ _i=1^nw_ix_ik, for any player i∈[n]i∈[n]. Then, the pure Nash Equilibrium of the multi-dimensional game is simply a cross-product of all d pureNE(Gk)pureNE(G_k) sets. More formally, a conversion between the Nash equilibrium in G and the 1D games GkG_k can be established through the following proposition: Proposition 10(Decomposition of Multi-dimensional AMG) Assume =∏k=1dk⊆ℝdX= _k=1^dX_k ^d, then the set of pure Nash equilibria of G can be characterized by pureNE(G)=∏k=1dpureNE(Gk)pureNE(G)= _k=1^dpureNE(G_k), where Gk=(n,k,ℓik)G_k=(n,X_k, _ik). Given this proposition, it is sufficient to form and solve the d one-dimensional games GkG_k independently, and take the Cartesian product of their pure Nash equilibrium sets to obtain pureNE(G)pureNE(G) for multi-dimensional AMG G. Example 11(NEs for Two-Player AMG in 1D and 2D) In a one-dimensional two-player AMG G with targets t1t_1 and t2t_2, pure actions x1,x2∈=[L,U]x_1,x_2 =[L,U], and mechanism x^=w0x0+w1x1+w2x2 x=w_0x_0+w_1x_1+w_2x_2, the pure NE set only depends on the relative positions of t1t_1 and t2t_2 with respect to the weighted middle point m, where m=w0x0+w1L+w2Um=w_0x_0+w_1L+w_2U if t1≤t2t_1≤ t_2, and m=w0x0+w1U+w2Lm=w_0x_0+w_1U+w_2L if t1>t2t_1>t_2. The closed form solution of pure NE can be divided into three cases: 1. If t1=t2=t_1=t_2=t: pureNE(G)=(L,L),ift<w0x0+L(w1+w2),(U,U),ift>w0x0+U(w1+w2),(x1,x2):x^=t,otherwise.pureNE(G)= cases\(L,L)\,&if\;t<w_0x_0+L(w_1+w_2),\\ \(U,U)\,&if\;t>w_0x_0+U(w_1+w_2),\\ \(x_1,x_2): x=t\,&otherwise. cases 2. If t1≠t2t_1≠ t_2 and t1<m<t2t_1<m<t_2 or t2<m<t1t_2<m<t_1: pureNE(G)=(L,U),ift1<t2,(U,L),ift2<t1.pureNE(G)= cases\(L,U)\,&if\;t_1<t_2,\\ \(U,L)\,&if\;t_2<t_1. cases 3. If t1≠t2t_1≠ t_2 and t1,t2≤mt_1,t_2≤ m or t1,t2≥mt_1,t_2≥ m: pureNE(G)=(L,max(L,1w2(t2−w0x0−w1L))),ift1<t2≤m,(max(L,1w1(t1−w0x0−w2L)),L),ift2<t1≤m,(U,min(U,1w2(t2−w0x0−w1U))),ifm≤t2<t1,(min(U,1w1(t1−w0x0−w2U)),U),ifm≤t1<t2.pureNE(G)= cases\(L, (L, 1w_2(t_2-w_0x_0-w_1L)))\,&if\;t_1<t_2≤ m,\\ \( (L, 1w_1(t_1-w_0x_0-w_2L)),L)\,&if\;t_2<t_1≤ m,\\ \(U, (U, 1w_2(t_2-w_0x_0-w_1U)))\,&if\;m≤ t_2<t_1,\\ \( (U, 1w_1(t_1-w_0x_0-w_2U)),U)\,&if\;m≤ t_1<t_2. cases In case 1, the two targets coincide. NE happens when x x is in the location closest to t. This is also the case when there can be infinite many pure NEs (the third branch). In case 2, the targets lie on different sides of m. Both players must exaggerate their actions to the maximum extent possible. In case 3, both targets lie on the same side of m. One player will exaggerate to the maximum extent, while the other either “wins the game” by forcing the mechanism to land on its own target or also exaggerates to the maximum when winning is impossible due to a large w0x0w_0x_0. We illustrate in Figure 1 the three NE cases for 1-dimensional two player AMG examples with domain =[L,U]X=[L,U], equal weights w1=w2=12w_1=w_2= 12, and bias term w0x0=0w_0x_0=0. In case 1, x1x_1 and x2x_2 can be anywhere in the gray box, as long as x x sits on t1=t2t_1=t_2. Thus, there are infinite pure NE. Case 2 and 3 both have a unique pure NE. In case 2 x1x_1 and x2x_2 both exaggerate to the extreme action, L or U, that is closest to their respective targets. In case 3, player 1 plays an interior action and “wins” as x x lands on t1t_1, while player 2 chooses an extreme action. x xt1t_1t2t_2x1x_1x2x_2Case 1x xt1t_1t2t_2x1x_1x2x_2Case 2x xt1t_1t2t_2x1x_1x2x_2Case 3 Figure 1: Pure NE of two-player 1D AMG with Equal Weights Now, for multi-dimensional games we can combine the 1D pure NEs. In particular, for a 2-dimensional game, the pure NE falls into six categories formed by combining the three 1-D characterizations, which yields 3×3=93× 3=9 cases, then eliminating the three mirroring cases. In Figure 2 we present these six characterizations of a 2-dimensional AMG with =[L,U]2X=[L,U]^2, uniform weights w1=w2=12w_1=w_2= 12, and w0x0=0w_0x_0=0. These derive directly from combining the three 1-dimensional NE cases in Figure 1. In case 1 (1=2t_1=t_2), 1,2x_1,x_2 can be anywhere in the gray box so long as they average to 1t_1. In cases 2 and 3, 1,2t_1,t_2 agree on one coordinate, and the infinite many pure NEs are chosen from the blue line and the green line, respectively. Cases 4,5,6 have a unique pure NE. Case 6 is where player 1 is the “winner” and has an interior action. x1x_12x_21t_12t_2Case 1 x1x_12x_21t_12t_2Case 2 x1x_12x_21t_12t_2Case 3 x1x_12x_21t_12t_2Case 4 x1x_12x_21t_12t_2Case 5 x1x_12x_21t_12t_2Case 6 Figure 2: Pure NE of two-player 2D AMG with Equal Weights 4 Bayesian Games under the Affine Mechanism In the previous formulation of AMG, we assumed that each player knows the target location 1,…,nt_1,…,t_n of all players, and hence all loss functions are common knowledge. However, it is also interesting to consider the scenario in which each player (say player i) only knows its own target it_i, but not any of the other players’ target −it_-i. For such games with incomplete information, we adopt an interim Bayesian game approach in which the player knows only the prior distributions of the other targets. Concretely, the Bayesian Affine Mechanism Game has the same formulation as the standard AMG, except that any target it_i shall be chosen from a distribution TiT_i, for all i∈[n]i∈[n]. Each player i knows the exact location of their own target it_i. They also know the distributions of TjT_j, ∀j∈[n]∀ j∈[n], the AMG mechanism, and that other players also know this information. The players then make the simultaneous action i∈x_i from the common continuous action space ⊂ℝdX ^d that is compact and convex. All other settings remain the same. We have the following definition: ℓi(i,−i|i) _i (x_i,x_-i|t_i ) ≔‖^−i‖22=‖w00+∑j=1nwjj−i‖22 \| x-t_i \|_2^2= \|w_0x_0+ _j=1^nw_jx_j-t_i \|_2^2 (5) Definition 12(Bayesian AMG) The Bayesian Affine Mechanism Game is an n-player general-sum Bayesian game G=(n,,Tii=1n,ℓii=1n)G= (n,X, \T_i \_i=1^n, \ _i \_i=1^n ), where n is the number of players, X is the compact and convex action space, TiT_i is the independent prior type distribution of player i’s target i∈ℝdt_i ^d and ℓi:n×ℝd→ℝ _i:X^n×R^d is given by (5). We denote a player’s strategy by i=πi(i)x_i= _i (t_i ) where πi:ℝd→ _i:R^d maps their target position to an action i∈x_i . We again approach the problem with the assumption of a product space domain =∏k=1d[Lk,Uk]X= _k=1^d[L_k,U_k], and the following result characterizes the best responses in the Bayesian game. Proposition 13(Bayesian Best Responses) The best response function for each player i in the Bayesian AMG game with =∏k=1d[Lk,Uk]X= _k=1^d[L_k,U_k] has the ramp function form, bri(π−i|ti)=i, where for every k∈[d] _i ( _-i|t_i )=x_i, where for every\;k∈[d] (6) xik=minUk,maxLk,1witik−1wi(w0x0k+∑j≠iwjTj[πjk(tj)]). x_ik= \U_k, \L_k, 1w_it_ik- 1w_i (w_0x_0k+ _j≠ iw_jE_T_j [ _jk (t_j ) ] ) \ \. In particular, (6) implies that in all Bayesian Nash equilibria, all players use a strategy with this ramp function form in all coordinates. To be specific, in a BNE, the ramp function can be rewritten as, for every player i, πi(ti)=i _i (t_i )=x_i, where xik=Lk if tik≤aik1witik+cik if aik<tik<bikUk if tik≥bik∀k∈[d]x_ik= casesL_k&\;if\;t_ik≤ a_ik\\ 1w_it_ik+c_ik&\;if\;a_ik<t_ik<b_ik\\ U_k&\;if\;t_ik≥ b_ik\\ cases\;∀ k∈[d], where cik=−1wi(w0x0k+∑j≠iwjTj[πjk(tj)])c_ik=- 1w_i (w_0x_0k+ _j≠ iw_jE_T_j [ _jk (t_j ) ] ), aik=wi(Lk−cik)a_ik=w_i (L_k-c_ik ), and bik=aik+wi(Uk−Lk)=wi(Uk−cik)b_ik=a_ik+w_i (U_k-L_k )=w_i (U_k-c_ik ). This set of equations in each dimension k characterizes ciki=1n \c_ik \_i=1^n thus πi _i. Notice that the equation set in each dimension can be calculated independently. This result can be interpreted as follows: in any dimension, the exaggeration factor is always 1wi 1w_i, that is, the less influence the player has, the more the player will exaggerate. For arbitrary prior type distributions, ciki=1n∀k∈[d] \c_ik \_i=1^n\;∀ k∈[d] can be solved numerically, but as an example, we can solve for a closed-form solution under the assumption of uniformly distributed targets for all players. Example 14(BNE for Uniformly Distributed Targets) A one-dimensional Bayesian AMG with =[0,1]X=[0,1], targets Ti∼Unif [0,1]T_i \;[0,1], bias term w0x0=0w_0x_0=0, and ∑i=1nwi=1 _i=1^nw_i=1 for all players has the unique BNE strategy profile (π1,…,πn)( _1,…, _n) given by πi∗(ti)=0,ti≤1−wi2,1witi−1−wi2wi,1−wi2<ti<1+wi2,1,ti≥1+wi2. _i^*(t_i)= cases0,&t_i≤ 1-w_i2,\\ 1w_it_i- 1-w_i2w_i,& 1-w_i2<t_i< 1+w_i2,\\ 1,&t_i≥ 1+w_i2. cases (7) tit_ixix_in=2n=2 uniform casen−12n n-12nn+12n n+12n011 tit_ixix_in=4n=4 uniform casen−12n n-12nn+12n n+12n011 tit_ixix_in→∞n→∞ uniform casen−12n,n+12n→12 n-12n, n+12n→ 12011 Figure 3: BNE of Games with n Players and Uniform Type Distributions (7) follows from Proposition 13 by solving the system of n equations in the form of (6). More details of the derivation can be found in the appendix. With w0=0w_0=0 and wi=1nw_i= 1n for every i, (7) can be further simplified and plotted for various values of n in Figure 3. In this example, 1−(bi−ai)=1−wi1-(b_i-a_i)=1-w_i fraction of the types of player i will report one of two extreme types 0 or 11, and the remaining wiw_i fraction will exaggerate by a factor of 1wi 1w_i. 5 Extensions of the Affine Mechanism Game 5.1 Extension 1: Negative Inner Product Loss Up to now player i’s loss function (2) is based on the Euclidean distance between its target point i∈ℝdt_i ^d and the mechanism x. In some applications, the following negative inner product loss can be more appropriate: ℓi():=−i⊤^. _i(x):=-t_i x. (8) That is, player i has a small loss if the mechanism x has a large projection onto the direction of target direction it_i. AMG with this negative inner product loss (8) has an even stronger guarantee: the game has a Weakly Dominant Strategy Equilibrium. Definition 15(Weakly Dominant Strategy Equilibrium (wDSE)) An action profile =(1…n)∈nx=(x_1…x_n) ^n is a wDSE if for every player i, ℓi(i,−i′)≤ℓi(,−i′),∀∈,−i′∈n−1 _i (x_i,x _-i )≤ _i (y,x _-i ),\; ,x _-i ^n-1. Remark 16 The term weakly dominant strategy equilibrium is used in Chapter 4.5 of Tadelis [2013], and it is also called dominant strategy equilibrium in Chapter 10.3 of Osborne [1994], and dominant strategy solution in Chapter 1.3.1 of Roughgarden [2010]. As noted in Osborne [1994], an action in a wDSE is not required to weakly dominate (with at least one strict inequality) all other actions for a player. wDSE is also a weaker solution concept compared to (strictly) dominant strategy equilibrium, which requires strict inequality everywhere, ℓi(i,−i′)<ℓi(,−i′),∀∈,≠i,−i′∈n−1 _i (x_i,x _-i )< _i (y,x _-i ),\; ,y _i,x _-i ^n-1. Proposition 17(Existence of wDSE) Under loss (8), game G has a wDSE i∗x_i^* satisfying i∗∈argmaxi∈i⊤i,∀i∈[n]x_i^*∈ argmax_x_i t_i x_i\;,∀ i∈[n]. One significant benefit of this wDSE is that each player can compute their i∗x_i^* on their own, without even the knowledge of other players’ actions j,∀j≠ix_j,∀ j≠ i. Contrast this with the best-response dynamics in section 3.1. We remark that i∗x_i^* may not be exactly along the direction of it_i since it depends on the domain X. Nonetheless, computing i∗x_i^* is a convex optimization problem–maximizing a linear function over a convex set–and thus efficient. 5.2 Extension 2: Finite Action Space So far, we have assumed that the players’ action space X is a compact and convex (hence infinite unless singleton) subset of ℝdR^d. In some applications, the players are restricted to picking their actions from a finite X instead. For example, X may be the collection of news articles (represented by embedding vectors in ℝdR^d) published by all professional news agencies within the past 24 hours, and each player may select a handful of such news articles to place on a social media user (the mechanism)’s timeline. This motivates the extension to finite action space: Definition 18(AMG with finite action space) Affine Mechanism Game with finite action space is an n-player general-sum game F=(n,ki()i=1n,ℓii=1n)F= (n, \P_k_i (X ) \_i=1^n, \ _i \_i=1^n ), where the action space of player i is ki()P_k_i (X ), and ki()P_k_i (X ) is the set of all subsets containing kik_i elements from a finite ⊂ℝdX ^d. The loss function of player i is given by ℓi()=‖^−i‖22 _i (x )= \| x-t_i \|^2_2 with ^=w00+∑i=1nwi∑j=1kii(j) x=w_0x_0+ _i=1^nw_i _j=1^k_ix (j )_i, where i(j)x^(j)_i is the jjth element in player i’s chosen subset. In the original AMG (Definition 1) where X is convex, allowing the players to choose multiple items would not affect the results since choosing multiple items is equivalent to choosing the mean of these items, which is still in X. In the new game, the average of items in X may not be in X. However, the new game can be interpreted as each player picks one “meta item” i′x _i instead of kik_i items, with wi′=wikiw _i=w_ik_i and i′=1ki∑j=1kii(j)x _i= 1k_i _j=1^k_ix (j )_i. Corollary 19 F is also a potential game, with a potential function ϕ(i(j)j=1kii=1n):=‖w00+∑i=1nwi′i′‖22−2∑i=1nwi′i⊤i′. φ ( \ \x (j )_i \_j=1^k_i \_i=1^n ):= \|w_0x_0+ _i=1^nw _ix _i \|_2^2-2 _i=1^nw _it _ix _i. (9) Therefore, the new game F also has at least one pure NE. Since X is finite and the number of players is finite, F is a finite game. Although a pure NE can be similarly found through best-response dynamics, unlike the continuous case, each iteration of best-response dynamics can be costly to compute for large values of kik_i since it involves solving a variant of the subset sum problem. Since F is finite, it clearly cannot have infinite pure NEs (contrast with corollary 6), but it can still have many pure NEs. 6 Conclusion and Future Work We introduced the Affine Mechanism Game and its Bayesian variant, and characterized the structures of pure Nash and Bayesian Nash equilibria, including closed-form solutions under simplifying assumptions. In particular, we showed the players’ extreme-exaggeration equilibrium behavior in the complete-information case and quantified the exaggeration ratio as a function of the players’ influence (weight). The presented work has several limitations that can be enhanced by future work, including broadening the mechanism to non-affine functions (such as trimmed mean) and characterizing the full set of NEs, and allowing uncertainty in the players’ beliefs about the mechanism (such as the weights wiw_i’s). 7 Technical Appendices and Supplementary Material 7.1 Proof of Theorem 3 Proof We need to show if any player i deviates from action ix_i to any action ∈y , we have ℓi(i,−i)−ℓi(,−i)=ϕ(i,−i)−ϕ(,−i). _i(x_i,x_-i)- _i(y,x_-i)=φ(x_i,x_-i)-φ(y,x_-i). To this end, define an auxiliary variable z that does not depend on ix_i or y: :=w00+∑j≠iwjj.z:=w_0x_0+ _j≠ iw_jx_j. Then ℓi(i,−i)=‖wii+(−i)‖2=‖wii‖2+2wii⊤(−i)+‖−i‖2 _i(x_i,x_-i)=\|w_ix_i+(z-t_i)\|^2=\|w_ix_i\|^2+2w_ix_i (z-t_i)+\|z-t_i\|^2, and ℓi(i,−i)−ℓi(,−i)=‖wii‖2+2wii⊤(−i)−‖wi‖2−2wi⊤(−i). _i(x_i,x_-i)- _i(y,x_-i)=\|w_ix_i\|^2+2w_ix_i (z-t_i)-\|w_iy\|^2-2w_iy (z-t_i). On the other hand, ϕ(i,−i) φ(x_i,x_-i) =‖wii+‖2−2wii⊤i−2∑j≠iwjj⊤j =\|w_ix_i+z\|^2-2w_it_i x_i-2 _j≠ iw_jt_j x_j =‖wii‖2+2wi(−i)⊤i+‖2−2∑j≠iwjj⊤j. =\|w_ix_i\|^2+2w_i(z-t_i) x_i+\|z\|^2-2 _j≠ iw_jt_j x_j. The last two terms do not depend on ix_i. Hence ϕ(i,−i)−ϕ(,−i) φ(x_i,x_-i)-φ(y,x_-i) =‖wii‖2+2wi(−i)⊤i−‖wi‖2−2wi(−i)⊤ =\|w_ix_i\|^2+2w_i(z-t_i) x_i-\|w_iy\|^2-2w_i(z-t_i) y =ℓi(i,−i)−ℓi(,−i). = _i(x_i,x_-i)- _i(y,x_-i). 7.2 Proof of Proposition 5 Proof Our potential function ϕφ is the sum of two convex functions in x and hence convex. Using utilities ui=−ℓiu_i=- _i, the game has concave potential −ϕ-φ, and since the domain nX^n is convex and ϕφ is smooth and convex, by Theorem 1 of Neyman [1997] and its corollary, the set of pure Nash equilibria coincides with the minima of the potential function on nX^n. 7.3 Proof of Corollary 6 Proof Since the set of minima of the convex potential function on a compact domain nX^n is non-empty, there is at least one pure Nash equilibrium. Since the set of minima of the convex potential function on a convex domain nX^n is convex (Corollary in Neyman [1997] below Theorem 1), any convex combination of two distinct pure Nash equilibria is another pure Nash equilibrium. 7.4 Proof of Theorem 7 Proof For any i∈[n]i∈[n], ∇iϕ=2wi∑k=0nwkk−2wii. _x_iφ=2w_i _k=0^nw_kx_k-2w_it_i. (10) Suppose ∃i,j∈[n],i≠j:i,j∈int∃ i,j∈[n],i≠ j:x_i,x_j \;X. Then ∇iϕ==∇jϕ⇒i=∑k=0nwkk=j, _x_iφ= 0= _x_jφ _i= _k=0^nw_kx_k=t_j, (11) a contradiction. The equation ^=i⋆ x=t_i follows from ∇i∗ϕ= _x_i^*φ= 0 and the definition of the mechanism (1). 7.5 Proof of Lemma 8 First we prove the following lemma: Lemma 20(Sorted Pure Nash Equilibrium) For a joint action x, if there exist i<ji<j such that xj<xix_j<x_i and ti<tjt_i<t_j, then x∉pureNE(G)x (G). Proof By the assumption in the lemma, at least one of the following must be true: Case 1: x^<tj x<t_j. Consider the unilateral deviation of player j to xj+ϵx_j+ε, for ϵ>0ε>0. Let x^′ x be the new mechanism, where x^′=x^+wjϵ x = x+w_jε. Choose ϵε small enough such that x^<x^′<tj x< x <t_j. This will always decrease the loss of player j. Case 2: x^>ti x>t_i. Similarly, player i unilaterally deviate to xi−ϵx_i-ε, for ϵ>0ε>0. The new mechanism is x^′=x^−wiϵ x = x-w_iε. Choose ϵε small enough such that x^>x^′>ti x> x >t_i. This will always decrease the loss of player i. In either case, some player has a unilateral deviation that strictly decreases their loss, so x∉pureNE(G)x (G). Now we actually prove Lemma 8 Proof Assume that the players are sorted in ascending order by their targets. By the Lemma 20, we know that in any pure NE, the players’ actions must be sorted by their targets. This shows that ∃i¯,i¯∈0,1,…,n+1∃ i, i∈\0,1,…,n+1\ s.t. xi=Lx_i=L for i≤i¯i≤ i, xi=Ux_i=U for i≥i¯i≥ i, because pure NE strategies of the players are sorted by their targets, and the action space is not unbounded. In other words, there could be a set of players playing L in the start of the queue of players sorted by their targets, and a potential set of players playing U in the end of that queue. Note that when i¯<1 i<1, we define that there will be no player playing L, and when i¯>n i>n, there will be no player playing U. In addition, by Theorem 7, ∀i∀ i s.t. i¯<i<i¯ i<i< i must satisfy ti=t⋆t_i=t for some t⋆∈[L,U]t ∈[L,U]. Therefore, xi=L∀i s.t. ti<t⋆x_i=L\;\;∀ i s.t. t_i<t , and xi=U∀i s.t. ti>t⋆x_i=U\;\;∀ i s.t. t_i>t . By Theorem 7, we have x^=t⋆ x=t , so there is x^=∑i=0nwixi=t⋆⟹∑i:ti=t⋆wixi=t⋆−w0x0−∑i:ti<t⋆wiL−∑i:ti>t⋆wiU x= _i=0^nw_ix_i=t _i:t_i=t w_ix_i=t -w_0x_0- _i:t_i<t w_iL- _i:t_i>t w_iU 7.6 Proof of Proposition 10 Proof The loss for any player i∈[n]i∈[n] in G can be separated this way: ℓi()=‖w00+∑j=1nwjj−i‖22=∑k=1d(w00k+∑j=1nwjjk−ik)2=∑k=1dℓik(k). _i(x)=\|w_0x_0+ _j=1^nw_jx_j-t_i\|_2^2= _k=1^d(w_0x_0k+ _j=1^nw_jx_jk-t_ik)^2= _k=1^d _ik(x_k). We argue that x is a pure Nash equilibrium if and only if, for each coordinate k∈[d]k∈[d], the profile (1k,2k,…,nk)(x_1k,x_2k,…,x_nk) is a pure Nash equilibrium of GkG_k. Assuming that the action profile x is a pure NE. Then ∀i∈[n],∀i′≠i,ℓi(i,−i)≤ℓi(i′,−i)∀ i∈[n],\; _i _i,\; _i(x_i,x_-i)≤ _i(x _i,x_-i). Suppose that ∃k∈[d],i∈[n],ik′≠iks.t.ℓik(ik,−ik)>ℓik(ik′,−ik)∃ k∈[d],\;i∈[n],\;x _ik _ik\;s.t.\; _ik(x_ik,x_-ik)> _ik(x _ik,x_-ik). Define a full-dimensional deviation ′x by changing only the kkth coordinate: xi′=(xi1,…,xi,k−1,xik′,xi,k+1,…,xid).x_i =(x_i1,…,x_i,k-1,x _ik,x_i,k+1,…,x_id). then the pure NE assumption is violated: ℓi(i,−i)=∑k=1dℓik(ik,−ik)>ℓi1(1)+⋯+ℓik(ik′,−ik)+⋯+ℓid(d)=ℓi(i′,−i). _i(x_i,x_-i)= _k=1^d _ik(x_ik,x_-ik)> _i1(x_1)+…+ _ik(x _ik,x_-ik)+…+ _id(x_d)= _i(x _i,x_-i). Hence, if x is a pure NE, kx_k is a pure NE for GkG_k. Conversely, assume that for all k∈[d]k∈[d], k=(ik,−ik)x_k=(x_ik,x_-ik) is a pure NE of GkG_k. Then ∀i,k,∀ik′∈k∀ i,k, _ik _k, ℓik(ik,−ik)≤ℓik(ik′,−ik) _ik(x_ik,x_-ik)≤ _ik(x _ik,x_-ik). Summing the LHS and RHS of the inequalities gives: ℓi(i,−i)=∑k=1dℓik(ik,−ik)≤∑k=1dℓik(ik′,−ik)=ℓi(i′,−i)∀i′≠i. _i(x_i,x_-i)= _k=1^d _ik(x_ik,x_-ik)≤ _k=1^d _ik(x _ik,x_-ik)= _i(x _i,x_-i) _i _i. This is exactly the definition of pure NE for G. Hence, if kx_k is a pure NE for GkG_k, x, which is created by stacking up all kx_k, is a pure NE for G. Hence we have pureNE(G)=∏k=1dpureNE(Gk)pureNE(G)= _k=1^dpureNE(G_k). 7.7 Proof of Proposition 13 Proof First, the expected loss of player i is separable: T−i[ℓi(i,π−i;i)] _T_-i [ _i(x_i, _-i;t_i) ] =T−i[∑k=1d(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2|i] =E_T_-i\! [ _k=1^d (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 |t_i ] =∑k=1dT−i[(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2|i]. = _k=1^dE_T_-i\! [ (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 |t_i ]. Fixing any π−i(−i) _-i(t_-i), the best response of player i given target it_i is: i∗ _i^* =argmini∈∑k=1dT−i[(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2|i] = _x_i _k=1^dE_T_-i [ (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 |t_i ] (12) When minimizing T−i[ℓi(i,π−i;i)]E_T_-i [ _i(x_i, _-i;t_i) ], term k of (12) depends on player i’s action only through xikx_ik, i.e. any change on xik′,k′≠kx_ik ,\;\;k ≠ k will not change term k. In addition, the domain is a product-space =∏k=1dkX= _k=1^dX_k, so the choice of xikx_ik is unaffected by the choice in any other coordinate. Thus, minimizing T−i[ℓi(i,π−i;i)]E_T_-i [ _i(x_i, _-i;t_i) ] is equivalent to minimizing each of its d terms. The distribution of targets T1,…,TnT_1,…,T_n are independent across players, so the expected action for each opponent is independent of it_i, which means Tj[πjk(j)∣i]=Tj[πjk(j)]∀j≠iE_T_j[ _jk(t_j) _i]=E_T_j[ _jk(t_j)]\;\;∀ j≠ i, given i and k. We will use this to evaluate the derivative below. Since term k is strictly convex in xikx_ik, its minimum on the domain [Lk,Uk][L_k,U_k] occurs at xik=Lkx_ik=L_k if and only if it is monotonically increasing on [Lk,Uk][L_k,U_k], which happens when dxikT−i[(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2]|xik=Lk≥0 ddx_ikE_T_-i [ (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 ] |_x_ik=L_k≥ 0 ⟹tik≤w0x0k+wiLk+∑j≠iwjTj[πjk(j)]. t_ik≤ w_0x_0k+w_iL_k+ _j≠ iw_jE_T_j [ _jk(t_j) ]. Similarly, the minimum on [Lk,Uk][L_k,U_k] occurs at xik=Ukx_ik=U_k if and only if term k is monotonically decreasing on [Lk,Uk][L_k,U_k], which happens when dxikT−i[(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2]|xik=Uk≤0 ddx_ikE_T_-i [ (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 ] |_x_ik=U_k≤ 0 ⟹tik≥w0x0k+wiUk+∑j≠iwjTj[πjk(j)]. t_ik≥ w_0x_0k+w_iU_k+ _j≠ iw_jE_T_j [ _jk(t_j) ]. The best response is an interior point of [Lk,Uk][L_k,U_k] when neither condition above holds, i.e. when term k of (12) is non-monotonic on [Lk,Uk][L_k,U_k]. It happens when its unconstrained global minimizer xik∗x_ik^* lies strictly within (Lk,Uk)(L_k,U_k), i.e. Lk<xik∗<UkL_k<x^*_ik<U_k. The unconstrained best response xik∗x_ik^* is defined as xik∗=argminxik∈ℝT−i[(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2].x_ik^*= _x_ik E_T_-i [ (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 ]. To find xik∗x_ik^*, we set the derivative of that expectation to 0, and we have: dxikT−i[(w0x0k+wixik+∑j≠iwjπjk(j)−tik)2]|xik=xik∗=0 ddx_ikE_T_-i [ (w_0x_0k+w_ix_ik+ _j≠ iw_j _jk(t_j)-t_ik )^2 ] |_x_ik=x^*_ik=0 ⟹xik∗=1witik−1wi(w0x0k+∑j≠iwjTj[πjk(j)]). x_ik^*= 1w_it_ik- 1w_i (w_0x_0k+ _j≠ iw_jE_T_j [ _jk(t_j) ] ). Therefore, term k is minimized at πik∗(i)=xik∗=Lk,tik≤aik,1witik+cik,aik<tik<bik,Uk,tik≥bik, _ik^*(t_i)=x_ik^*= casesL_k,&t_ik≤ a_ik,\\ 1w_it_ik+c_ik,&a_ik<t_ik<b_ik,\\ U_k,&t_ik≥ b_ik, cases where cik=−1wi(w0x0k+∑j≠iwjTj[πjk(tj)])c_ik=- 1w_i (w_0x_0k+ _j≠ iw_jE_T_j [ _jk (t_j ) ] ), aik=wi(Lk−cik)a_ik=w_i (L_k-c_ik ), and bik=aik+wi(Uk−Lk)=wi(Uk−cik)b_ik=a_ik+w_i (U_k-L_k )=w_i (U_k-c_ik ). 7.8 Proof of Example 14 Proof By Proposition 13, the BNE set consists of ramp profiles parameterized by (c1,…,cn)(c_1,…,c_n) that satisfy the consistency system on our only dimension. We will solve for ci∀i∈[n]c_i\;∀ i∈[n]. First, since 0≤Tj[πj(tj)]≤1∀j,πj0 _T_j [ _j (t_j ) ]≤ 1\;∀ j, _j, for any BNE strategy profile πj∗(tj)j=1n\π^*_j(t_j)\_j=1^n, we have the following properties for ai,bia_i,b_i of player i: ai a_i =∑j≠iwjTj[πj∗(tj)]≥∑j≠iwj⋅0=0 = _j≠ iw_jE_T_j [π^*_j (t_j ) ]≥ _j≠ iw_j· 0=0 bi b_i =wi+∑j≠iwjTj[πj∗(tj)]≤wi+∑j≠iwj=1 =w_i+ _j≠ iw_jE_T_j [π^*_j (t_j ) ]≤ w_i+ _j≠ iw_j=1 This means for strategy πi∗(ti)π^*_i(t_i) in any BNE, it must satisfy 0≤ai<bi≤10≤ a_i<b_i≤ 1. We are not limiting the action space of the game, because this constraint is directly derived from existing constraints, including =[0,1]X=[0,1] and the characterizations of aia_i and bib_i. Therefore we can compute Tj[πj∗(Tj)]E_T_j [π^*_j(T_j) ] under the uniform prior by assuming that 0≤ai<bi≤10≤ a_i<b_i≤ 1 holds for this BNE strategy: Tj[πj∗(tj)]=0⋅aj+1⋅(1−bj)+∫ajaj+wj(1wjtj−ajwj)tj=1+wjcj−wj2.E_T_j [π^*_j(t_j) ]=0· a_j+1·(1-b_j)+ _a_j^a_j+w_j\!\! ( 1w_jt_j- a_jw_j )dt_j=1+w_jc_j- w_j2. Substituting into the consistency system ci=−1wi∑j≠iwjTj[πj∗(tj)]c_i=- 1w_i _j≠ iw_jE_T_j [π^*_j (t_j ) ] yields the linear system wici+∑j≠iwj2cj=−∑j≠i(wj−wj22),∀i∈[n].w_ic_i+ _j≠ iw_j^2c_j=- _j≠ i (w_j- w_j^22 ), ∀ i∈[n]. Let the matrix representing the LHS weights be M, with Mij=wiifj=iwj2ifj≠iM_ij= casesw_i&if\;j=i\\ w_j^2&if\;j≠ i cases subtracting row 11 from all other rows, and then applying row 1−wi2wi−wi2⋅rowi,∀i∈2,…,nrow\;1- w_i^2w_i-w_i^2·row\;i,\;∀ i∈\2,…,n\ makes a lower triangular matrix with non-zero diagonal values. This means that M is full rank and there is a unique solution to our system of equations. ci=−1−wi2wi∀i∈[n]c_i=- 1-w_i2w_i\;∀ i∈[n] is the unique solution here, with verification omitted. If we plug cii=1n\c_i\_i=1^n back into the ramp function, the unique BNE strategy has ai=1−wi2,bi=1+wi2a_i= 1-w_i2,b_i= 1+w_i2, and ci=−1−wi2wic_i=- 1-w_i2w_i, which matches our claim in Example 14. 7.9 Proof of Proposition 17 Proof Given an arbitrary joint action from other players −ix_-i, player i’s best response is BR(−i) BR(x_-i) :=argmini∈ℓi(i,−i)=argmini∈−i⊤(w00+∑j=1nwjj) := argmin_x_i _i(x_i,x_-i)= argmin_x_i -t_i (w_0x_0+ _j=1^nw_jx_j) =argmini∈−wii⊤i+const=i∗. = argmin_x_i -w_it_i x_i+const=x_i^*. The best response is independent of −ix_-i. 7.10 Proof of Corollary 19 Proof The proof is identical to the one for Theorem 3, since the loss functions are identical, only on a different domain. References Z. Barak, A. Gupta, and I. Talgam-Cohen (2024) MAC advice for facility location mechanism design. Advances in Neural Information Processing Systems 37, p. 129564–129604. Cited by: §2. D. Berga (2002) Single-peakedness and strategy-proofness of generalized median voter schemes. 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