Mapping an Information Design Game into an All-Pay Auction

I formally establish the existence of a mapping between a class of information design games with multiple senders and a class of all-pay auctions. I fully characterize this mapping and show how to use it to ﬁnd equilibria in the information design game. The mapping allows for a straightforward comparative statics analysis of equilibria in the latter class of games. I use it to study the eﬀect of the tie-breaking rule on the distributions of posteriors and the receiver’s payoﬀ.


Introduction
Games of information design typically involve an uninformed sender who chooses an experiment structure that induces the receiver to behave in a manner that the sender finds desirable.These games have received considerable attention in the theoretical literature in the past decade.A large part of this literature relaxes or modifies the assumptions of the baseline model of Kamenica and Gentzkow (2011).While most of the articles in this literature focus on the single sender, some consider multiple senders (Gentzkow and Kamenica (2016), Boleslavsky and Cotton (2018), Au and Kawai (2019), Au and Kawai (2020)).There, the senders effectively compete for the receiver's attention.Of particular interest for this paper is the article by Boleslavsky and Cotton (2018).It studies two entrepreneurs who, through the choice of an experiment, are persuading an investor to choose their project over a competitor.Using that model, in this paper we establish a mapping between a class of information design games with multiple senders and a class of contests, particularly all-pay auctions.
Contests and all-pay auctions have been studied extensively, and there exists a significant body of results and understanding of these games.Baye et al. (1996) were the first to fully characterize the equilibria in the all-pay auction with multiple bidders.Clark and Riis (1998) studies the case of multiple prizes.Siegel (2009) further develops our understanding of such games by considering asymmetries between the players and allowing for a general class of cost functions.
All-pay auctions have also been widely applied to study competitive environments, for instance, lobbying and campaign spending (Hillman and Riley (1989), Baye et al. (1993), Che and Gale (1998), Sahuguet and Persico (2006)) and patent and R&D races (Moldovanu and Sela (2003), Che and Gale (2003)).The current understanding of all-pay auctions allows us to apply these games to study information design problems.In this paper, after establishing the mapping between the information design game and the all-pay auction, summarized in Figure 4, we employ the equilibria of the all-pay auction to characterize the equilibrium behavior in the information design game.Further, using the auxiliary "valuations" variables (V 1 , V 2 ), determined as part of the mapping, we conduct a comparative statics analysis with respect to the tie-breaking rule. 1 The latter analysis allows for determining the receiver-preferred tie-breaking rule.
The approach used in this article, of first finding the correspondence between the two classes of games and then using the results from contests to study information design, could be applied to other similar games or generalized.In particular, the results from all-pay auctions with multiple asymmetric bidders can be applied to study the generalization of Boleslavsky and Cotton (2018) with three and more asymmetric projects.Hence, there is potentially a significant scope for applying the results from all-pay auctions and contests literature to current problems in information design.

Setup
Consider a model of two entrepreneurs (senders) competing for the funds of a single investor (receiver).The investor has enough money to invest in one indivisible project only.Each entrepreneur has one project idea, good or bad.The qualities of the two projects are independent.Let the ex-ante probability that entrepreneur i has a good project be α i,0 ∈ (0, 1).
A project requires an investment of r ∈ (0, 1).The good project brings a gross return of 1 to the investor, and the bad project brings 0. If the investor chooses to invest in the project of entrepreneur i, that entrepreneur gets a fixed payoff of w > 0; otherwise, he gets 0. The investor may choose not to invest in any project.
The true qualities of the projects are unknown to any player.The information is symmetric throughout the game.Before the investor chooses the project to invest in (if any), the entrepreneurs simultaneously conduct informative experiments about the qualities of their respective projects.The results of these experiments are public.
It is without loss to formalize the entrepreneur i's choice of the experiment as a choice of the distribution of posterior beliefs about the quality of i's project, considerably.

Characterizing the mapping
Applying a technique similar to the one used in the appendix of Sahuguet and Persico (2006), consider entrepreneur i's decision when choosing the distribution of posteriors, G i , and fix i's opponent's distribution of posteriors, G k .Entrepreneur i's optimized payoff at this stage is , and G i is a CDF.
Writing down the Lagrangian that corresponds to this optimization problem, we have After a series of transformations, we can also write the Lagrangian down as Consider the integrand of the above expression and, in turn, rewrite it as an integral with respect to the opponent's CDF as a measure: Without using the indicator functions, write down the latter integrand as Recall that player i controls the distribution of x and player k -that of t.In the above expression, we can re-interpret x as player i's choice of a bid, b i , and t as his opponent's choice of a bid, b k .Note also that having a higher bid is necessary for winning the prize of value w λ i , that the bid always has to be paid, regardless of winning or losing, and that one can only ever win by bidding above r but below 1.We can conclude that this expression coincides with the payoff of a contestant in the all-pay auction with a reserve price r, a bid cap of 1, a tie-breaking rule Note that in the corresponding all-pay auction, the equilibrium pair of expenditures equals (α 1,0 , α 2,0 ).Let us characterize equilibria in the all-pay auction for all pairs of valuations (V 1 , V 2 ) that result in expenditures falling within the unit square, (α 1,0 , α 2,0 ) ∈ (0, 1) 2 .Doing so allows us to characterize the equilibria in the original information design game.Setting the pair of equilibrium expenditures in the all-pay auction game equal to the pair of prior probabilities in the original information design game would then pinpoint the exact mapping.

Equilibrium Bidding and Expenditures
Denote player i's valuation from winning the item as V i .In a working paper, Muratov (2021), I find and characterize the equilibria of the all-pay auction with the reserve price and the bid cap.3 Figure 1 shows the parameter regions with different equilibrium regimes.
Below is a brief description of the equilibrium behavior in each zone: 2a shows the support of equilibrium bidding strategies and the atoms.The circles indicate the atoms, and the solid lines indicate the support of continuous bidding.
Note that bidder 1 has multiple equilibrium strategies in zone A , as t ∈ [0, 1].
t is a free parameter, with t×r V 2 standing for the size of the atom that player 1 has at bidding r; Figure 2b plots a typical pair of CDFs for this case.
(B, B ) Zones B and B are defined as Figure 3a shows the equilibrium bidding in those regions. 4.Figure 3b shows a pair of CDFs for a typical value of (V 1 , V 2 ) in the region B.
Recall that the prior expected qualities of projects in the information design game correspond to the expected spending of the players in the all-pay auction.
Figure 4 shows the correspondence between geometric regions of different types of equilibria in the space of prize valuations, (V 1 , V 2 ), and in the space of priors/expenses, (α 1,0 , α 2,0 ).
Several observations are worth pointing out.
First, notice that zones A and B are line segments in the (V 1 , V 2 )-space; thus, corresponding equilibria occur in knife-edge cases.However, in the (α 1,0 , α 2,0 )space, the same zones have non-zero measures.This phenomenon occurs precisely because of the multiplicity of equilibria in A and B of the all-pay auction game.
Namely, various sizes of player 1's atom at r lead to multiple levels of α 1,0 possible for a fixed value of α 2,0 .
Second, note that regions C and D have a non-empty intersection in the -space, while in the (α 1,0 , α 2,0 )-space, they only have a knife-edge intersection along α 2,0 = 1−r 1−rρ 2 .This happens because C and D in the all-pay auction lead to different equilibrium bidding and, as a result, different ranges of expenditures, which correspond to non-overlapping (except for the line segment) C and D in the information design game.
The expressions for the geometric regions of different equilibrium regimes in the (α 1,0 , α 2,0 )-space are: , where In the expressions above, ψ being properly defined is ensured by 2 .It is also useful to define how the priors (α 1,0 , α 2,0 ) map into the valuations in zones with continuous bidding, i.e.A, A , B, and B : We can now state the mapping result: Proposition 1.There exists a unique mapping between equilibria in the information design game and the all-pay auction.For every pair of priors in zones A-D of (α 1,0 , α 2,0 )-space, this mapping defines a pair of valuations in a corresponding zone of (V 1 , V 2 )-space.For each zone, the equilibrium distributions of posteriors in the information design game are the same as those in the all-pay auction.The mapping is defined by 3.1.1-3.1.8and 3.2.1-3.2.4.
We derive the formulae for valuations and the correspondence between the geometric regions of priors and valuations in Appendix A.
The algorithm to find the equilibrium in the information design game using the mapping is the following.Given an exogenous pair of priors, (α 1,0 , α 2,0 ), and fixing the equilibrium strategy of the investor, summarized by the tie-breaking rule, (ρ 1 , ρ 2 ), we can determine in which region, A to D, the pair of priors fall.
Then, depending on the region, we know what type of equilibrium behavior the bidders follow.The formulae provide the pairs of valuations (V 1 , V 2 ) for each pair of priors.Having the valuations, we can write down the expressions for the equilibrium CDFs.
Using this mapping, we can also perform the comparative statics exercise.
Having the auxiliary valuations variables is especially useful for that purpose.
From the point of view of the information design game, those variables are endogenous.However, in the all-pay auction game, they are exogenous, and the way the distributions react to changes in valuations is well understood.
Investor's payoff The first-best investor's payoff occurs under the perfectly informative experiment.In equilibrium, it occurs in the zone C. The support of posteriors there is {0, 1} 2 .In other regions, there are some informational losses.
It is important to notice that in regions A and B , if α 1,0 increases, but the pair of priors stay within the same region, it only results in entrepreneur 1 putting a higher mass on atom at r. Hence, such an increase in α 1,0 does not lead to higher investor's payoff.However, if both priors increase along the 45-degree line in A , or along the α 1 (α 2,0 , ρ 1 , ρ 2 )-line in B , it leads to changes in the supports of posteriors and increases the investor's payoff.Moreover, along the 45-degree line in A and along the α 1 -line in B , entrepreneur 1 does not have an atom at r, but has an atom at 0. There is also some continuously distributed mass above r but below 1.In zone B , as priors approach the zone C along the α 1 -line, the continuous parts of distributions shrink, and the supports become increasingly closer to {0, 1} 2 , unlike in zones B and D, where entrepreneur 1 always has an atom at r.

Tie-breaking rule analysis
In this section, we analyze the effect of varying the tie-breaking rule (ρ 1 , ρ 2 ) in the information design game.Here, we treat the valuations in the all-pay auction preimage game, (V 1 , V 2 ), as endogenous, and the prior beliefs in the information design game, (α 1,0 , α 2,0 ), as exogenous.
In the zones A and A , the tie-breaking rule plays no role.In the zones B, B , and D, the tie-breaking rule has two effects: it shifts the borders of those regions, and it changes the equilibrium behavior of the entrepreneurs.Besides, the tie-breaking rule also affects the borders of C.
In the series of lemmata below, we first characterize how the entrepreneurs' behavior changes with the tie-breaking rule within the zones, and then how the borders of the zones change.
The investor prefers the equilibrium under q ρ to that under ρ.
The proof of the lemma above is in Appendix B.1, where we apply the implicit function theorem to the system , and V 2 is increasing.After that, knowing how the CDFs depend on V 1 and V 2 , we can conclude the stochastic ordering stated in the lemma.
Recall also that experiment A is Blackwell more informative than experiment B if the distribution of posteriors under B dominates that under A in the sense of SOSD Since both distributions of posteriors are dominated for the lower value of ρ 1 , they are more informative, and the receiver prefers more informative outcomes.
We plot the change of distributions with respect to an increase of ρ 1 in zone Continue with the comparative statics with respect to ρ in the zone B .
The proof in Appendix B.2 again relies on applying the implicit function theorem to the system that defines expenditures in zone B joint with the equation Note that the increase of ρ 1 has opposite effects on the investor's payoff in zones B and B : the payoff increases with ρ (i) Upper bound of B on α 1,0 increases, and upper bound on α 2,0 decreases; (ii) Upper bound of B on α 2,0 decreases, and both bounds on α 1,0 increase; (iii) Upper bound of D on α 2,0 decreases, and lower bound on α 1,0 increases.
We prove this lemma in Appendix B.3.What Lemma 3 also says is that 7 depicts a zones shift with respect to an increase in ρ 1 .It would also be helpful to study how C changes, varying the tie-breaking rule, and find the range of priors that C can span.From the definition of C, it is easy to see that with the increase of ρ 1 , the lower bound on α 1,0 increases, while the lower bound on α 2,0 decreases.
In order to determine the frontier of points spanned by C, consider the point . It lies at the intersection of C's lower bounds on α 1,0 and α 2,0 .Note that for any such point, it holds that .
We can now state the result regarding C: Figure 8 depicts the effect of ρ 1 on C together with the α f rontier 1,0 .

Investor-preferred tie-breaking rule
With the above comparative statics results at hand, let us analyze the tie-breaking rules that maximize the investor's expected payoff.In this section, we focus on the case of α 1,0 α 2,0 .7  the investor choose the tie-breaking probabilities?Without being too formal, we could allow the investor to make announcements about the intended tie-breaking before the entrepreneurs choose their experiments.Unlike the announcements in the form of "only projects with posterior equal to 1 will be funded," statements about tie-breaking are cheap to make and follow through since they do not require commitment.We can then focus on equilibria such that entrepreneurs believe the investor and the investor makes truthful announcements.
Call the set of priors for which the first-best investor's payoff is achievable C.
Let us now determine the preferred tie-breaking rule when the investor's firstbest is not achievable.We will focus on the case of α 1,0 α 2,0 , with the remainder being symmetric.
Note that neither the borders of zones A or A , nor the distributions of posteriors there depend on ρ.So, we can further exclude from the analysis the region .
Naturally, such a tie-breaking rule allows the investor to achieve a local maximum.The other two types of tie-breaking rules which can achieve local maxima are ("rule 0") and ("rule 1") The former rule, "rule 0", either moves a prior into zone D (for α 1,0 1 − r + r 2 ) or zone B with ρ 1 = 0 (for α 1,0 < 1 − r + r 2 ).In zone B, the investor's payoff decreases with ρ 1 , so setting it as low as possible within zone B is desirable.In Appendix C we characterize an equilibrium in B under ρ 1 = 0 Following the "rule 1" achieves the same payoff of α 1,0 + α 2,0 − α 1,0 α 2,0 − r when the prior can move to zones B M or D M (i.e. when α 2,0 1+r 2 2 ).We summarize these observations in the proposition below with the formal analysis in Appendix D.

2
; in panel (b), -to region ; in panels (c) and (d), priors belong to region and in panels (e) and (f), -to region 1+r 2 2 , 1 × 1+r 2 2 , 1 .Furthermore, Figure 12 shows which of the tie-breaking rules achieve local maximum depending on the parameter region.
Finally, let us establish the conditions for which the investor prefers the internal tie-breaking rule ρ to the "corner" rules, i.e. "0 rule" and "1 rule".Note that for every pair of priors (α 1,0 , α 2,0 ) ∈ [ 1−r 2 2 , 1) 2 \C, we can find ρ that places the prior on the border between B and B M .Note also that in this case both entrepreneurs have atoms at {0} ∪ {1}, some continuously distributed mass of posteriors in (r, α), but no atoms at r.If we fix an initial prior (α 1,0 , α 2,0 ) and corresponding ρ (α 1,0 , α 2,0 ), there exists a path along α 1 (α 2,0 , ρ 1 , 1 − ρ 1 ) towards the first-best region C. Along that path, the distributions of posteriors are ordered in the sense of First Order Stochastic Dominance: as α 2,0 increases, the equilibrium distributions increase.As priors get sufficiently close to C, the continuous parts of both distributions shrink and the distributions approach the perfectly informative and the investor's payoff approaches the first-best payoff On the other hand, for priors we consider under "0 rule" and "1 rule", the payoff remains fixed at α 1,0 + α 2,0 − α 1,0 α 2,0 − r.The latter payoff is strictly lower than the first-best payoff.There exist priors sufficiently close to C, such that the distributions of posteriors under (ρ 1 , 1 − ρ 1 ) are sufficiently close to the perfectly informative distributions and thus, the investor's profit is sufficiently close to the first-best payoff and is also higher than α 1,0 + α 2,0 − α 1,0 α 2,0 − r.We can conclude that for priors close to C, the investor prefers the internal tie-breaking rule (ρ 1 , 1 − ρ 1 ).
Corollary 2. If the pair of priors is close enough to C, the investor prefers tiebreaking rule (ρ 1 , 1 − ρ 1 ) to the corner "0 rule".
It is worthwile to mention that the investor might prefer the corner "0 rule" for the robustness reasons.While fine-tuning the ρ -rule requires the exact knowledge of the pair of priors and, if (α 1,0 , α 2,0 ) are misjudged even slightly, might result in profit losses.On the other hand, the local optimality of "0 rule" only requires for priors to be in a certain range, α 1,0 max{ 1+r 2 2 , α 2,0 }, (α 1,0 , α 2,0 ) / ∈ C. Besides, the "0 rule" might also be more attractive from the fairness perspective, since it discriminates against the initially stronger entrepreneur 1, unlike the ρ -rule, which discriminates against the weaker entrepreneur 2.

Discussion
The approach studied in this paper allows characterizing equilibria with arbitrary tie-breaking rules, which is a generalization of the analysis in Boleslavsky and Cotton (2018).There, they study an equilibrium with the symmetric tie-breaking rule only.Leveraging the generality of the analysis, we also extend the analysis to find the investor-preferred tie-breaking rule.Using the methods from Boleslavsky and Cotton (2018) to study a general tie-breaking rule would reduce tractability.
Similarly, the bounds of B on α 2,0 reduce to α 2,0 ∈ 0, α 2 1,0 − r 2 , as in zone A. Moreover, the expressions for values (V 1 , V 2 ) in zones B and B become equal to those in zones A and A , respectively.Hence, it is a manageable task to write down equilibrium distributions in terms of priors in zones B and B in case of ρ 1 = 1/2 (unlike in the case of general ρ 1 ∈ (0, 1)).
The equilibrium analysis under the symmetric tie-breaking rule, ρ 1 = ρ 2 = 1/2 in Boleslavsky and Cotton (2018) relies on the direct analysis of the Lagrangian as in expressions (L)-( L).After proving some general properties of equilibrium distributions (uniformly distributed continuous part and atoms only at 0, r, and 1), the analysis there proceeds with establishing the supports of strategies and zones, where different types of equilibria hold.A set of conditions necessary for equilibrium are used to identify the latter.In case of symmetric tie-breaking, the expressions that pin down the supports and equilibrium regimes are tractable.
However, allowing for a general tie-breaking would require working with algebraically more complex expressions, especially in the zones where there are atoms at the posterior 1.
Another complexity that our approach simplifies is the comparative statics of distributions.With general tie-breaking, the expressions of equilibrium CDFs in terms of priors and (ρ 1 , ρ 2 ) feature many non-linear terms.They do not allow to make unambiguous conclusions about the effect of parameters' changes.However, writing the expressions in terms of auxiliary values is straightforward and allows for tractable analysis.
Our approach can also be applied to the information design problem with three senders.In Muratov (2021), we characterize equilibria in the three player all-pay auction with a reserve and a bid cap, building on the work of Baye et al. (1996).We show that in equilibria, the supports of the three players' strategies take the following form: If there is continuous bidding, distributions are such that any one active player faces a uniform distribution of maximum of competitors' bids.Knowledge of this equilibrium structure can be used to characterize corresponding equilibria in the three-sender information design game.
In a related paper, Szech (2015) studies the role of tie-breaking rules in the all-pay auction with bid caps, which generalizes the analysis of Che and Gale (1998).After establishing the equilibrium under general bid cap and tie-breaking rule, Szech (2015) studies the principal's design problem, whose goal is to maximize the expected sum of bids.She concludes that it is optimal to favor the weaker bidder, and, if the choice is among the simple tie-breaking rules, to do so deterministically.Our analysis focuses on tie-breaking rules in the information design environment, where the principal prefers higher informativeness under fixed average total expenditures.Similarly to Szech ( 2015), deterministic favoring of the weaker player can be optimal, but not generally.Moreover, random favoring of the stronger player can outperform the latter rule.
In this article, we employ the knowledge of contests to better understand an information design game.There is also a growing literature that also combines the two disciplines, primarily focusing on applying information design in contests.For example, there are studies of optimal principal's disclosure policy in Tullock contests (Zhang and Zhou ( 2016)), all-pay auctions (Chen et al. ( 2017)), and binary action contests (Melo Ponce (2018)).Besides the player ability-types, articles also study disclosing information about number of contestants (Feng and Lu (2016)) and Tullock discrimination parameter (Feng (2020)).

Conclusion
In this paper we have formally established and characterized the mapping between a class of information design games and a class of all-pay auctions.We have shown that solving for the equilibria in the latter game is helpful in finding the equilibria in the former game.Building on the work of Boleslavsky and Cotton (2018), our approach allows for a tractable generalization to an arbitrary tie-breaking rule.
Our approach also allows for straightforward comparative statics.We apply it to study the effects of tie-breaking rule on equilibrium informativeness, and to determine the principal-preferred tie-breaking rule.We conclude that the investor benefits from a non-symmetric tie-breaking rule.Up to three types of tie-breaking rules can be locally optimal.Deterministically favoring the weaker sender can be optimal.However, favoring the stronger sender can also be optimal in certain cases.In general, following the approach presented in this paper, more classes of contests could be established as the correspondences for Information Design games in the future, which would allow applying well-established tools from the contest/all-pay auctions theory to the study of information design games.
and Jingfeng Lu, "The optimal disclosure policy in contests with stochastic

A Deriving the correspondences
We will go through the regions of valuations (V 1 , V 2 ) outlined in 3.2, and as we do so, we will state the distributions, the find the expressions for average expenditures, and then derive the mappings from expenditures to valuations, inverting the expressions.
Region A Following the derivations in Muratov ( 2021), in region A, bids are distributed according to the CDFs: The pair of expenditures is then Solving for the valuations in the above expressions and selecting the solution such that V 1 0, we have that Using the fact that in the region A, V 2 ∈ [r, 1) and V 1 ∈ (V 2 , +∞), and the expressions above, we can translate the boundaries of region A in terms of valuations into the terms of expenditures: in the unique equilibrium of the all-pay auction game, the players' bids are distributed according to the CDFs: > 0 (corresponds to the equilibrium payoff of player 1 in the all-pay auction game).
The average expenditures are Solving for valuations in terms of expenditures, and selecting such the solution that 1 V 2 1−ρ 1 r ρ 2 , we have , to plus infinity we get that in terms of expenditures, the region is translated into ria in the all-pay auction.In every equilibrium the players' bids are distributed according to the CDFs: where r ∈ [0, 1], q ∈ [0, 1], t × q = 0. Again, we focus on q = 0.
The pair of expenditures is then .
Solving for V 2 and selecting the solution that falls withing the proper range for V 2 , we get two expressions for V 2 , in terms of (α 1,0 , t) and α 2,0 : Varying V 2 from 1 to 1−ρ 1 r 1−ρ 1 , and t from 0 to 1, we express the region B as where there is an equilibrium in which the players bid according to bid Then the average expenditures are and the valuations in terms of expenditures are there is a class of equilibria in which the players bid according to bid where t ∈ [0, 1] is a free parameter.This translates into expenditures and valuations in terms of priors and varying V 2 from 1−rρ 1 ρ 2 to 1 ρ 2 and t from 0 to 1, we get that , there is an equilibrium, in which the players bid according to bid The expenditures are and valuations in terms of priors are As we vary V 1 and V 2 , we get that

B Comparative statics B.1 Proof of Lemma 1
Proof.Consider the system of equations that defines the expenditures in region and transform it to where we also use the fact that ρ 1 = 1 − ρ 2 .Call this system Ξ.Keeping the pair of expenditures/priors fixed, we treat V 1 and V 2 as implicit functions of ρ 1 , that follow from Ξ.We are interested in the signs of dV 1 dρ 1 and dV 2 dρ 1 .Denote the matrix of derivatives of Ξ with respect to (V 1 , V 2 ) T as J.Then, we have Keeping in mind that dρ 2 = −dρ 1 , the vector of derivatives of Ξ with respect to After plugging in the values for (α 1,0 , α 2,0 ) from (4.1.1)-(4.1.2) and some simplifications, we get that .
Let us first determine the sign of dV 1 dρ 1 .The denominator of that expression is always positive.In the numerator, the term 2ρ as a function of V 2 it achieves a local maximum at and a local minimum at Checking the values of ) at the extremes of V 2 , i.e V 2 = 1 and is indeed negative.This allows to conclude that, overall, dV 1 dρ 1 is negative in B and strictly so for V 2 > 1.
Let us now determine the sign of dV 2 dρ 1 .The numerator of the expression for dV 2 dρ 1 is always positive.The denominator is increasing in V 2 .For V 2 = 1, the value of the denominator is (1 − r 2 )ρ 2 1 > 0, hence the denominator is positive, and, overall, Having determined the derivatives of V 1 and V 2 with respect to ρ 1 , we are now ready to make conclusions about the stochastic ordering of distributions of posteriors with respect to tie-breaking rules.Recall that for the continuous part of distribution, the slope of G 1 (α) is given by 1/V 2 .Thus, with an increase of ρ 1 , V 2 increases and the slope of G 1 (α) decreases.Let q G . It is necessary for G 1 and q G 1 to intersect exactly once in (r, 1), since they correspond to random variables with the same expectation, α 1,0 .Thus, the supremum of continuous support corresponding to G 1 must be higher, 1−q ρ 2 q These observations allow to conclude that ∃!α † ∈ (r, 1) such that for posteriors less than α † , G 1 (α) q G 1 (α); for posteriors greater with strict inequality for all α ∈ (r, 1).Therefore, G 1 SOSD q G 1 , and the increase of ρ 1 in the zone B increases 1's distribution in the SOSD-sense.
Continue with the analysis of G 2 .Recall that the slope of G 2 in the continuous . Since the corresponding distributions have equal expectations, it is necessary for G 2 and q G 2 to intersect exactly once in (r, 1).In order for the latter to occur, it must hold that G 2 (x) < q G 2 (x), x < r.Hence, is the value of G 2 (x) for x ∈ [0, r).Again, we conclude that there exists a unique α ‡ ∈ (r, 1): with strict inequality for all α ∈ (r, 1), i.e.G 2 SOSD q G 2 .
Consider now the preferences of the investor.For a fixed realization of posteriors, (α 1 , α2 ) = (x, y), the payoff of the investor is v where by [.] + we denote the positive part.The expected payoff with respect to distributions of posteriors is, therefore, Note that v P (x, y) = [max{x, y} − r] + is a convex function of x, while a system, which we will denoted as Ξ .As in the previous proof, use the implicit function approach to determine the signs of dV 1 dρ 1 and dV 2 dρ 1 that follow from the system above.Treat the variables (V 1 , V 2 , t) as endogenous, and ρ 1 and ρ 2 = 1 − ρ 1 as exogenous.The matrix of derivatives of the system above with respect to , with the inverse .
Let us determine the sign of dV 1 dρ 1 .It is straightforward to check that the numerator is negative, given V 2 1.The denominator in terms of V 2 is a parabola with a minimum at 1, the denominator is increasing.Check that at V 2 = 1 the sign of it is positive to conclude that dV 1 dρ 1 is negative in the zone B .Similarly, dV 2 dρ 1 is positive in B .We are now ready to conduct the comparative statics of G 1 and G 2 , starting with the latter.Let ρ 1 > q ρ 1 .Recall that in B , r/V 1 is the value of G 2 (x) for x ∈ [0, r], and 1/V 1 is its slope in x ∈ [r, (1 for all x ∈ [0, α ], where α > r is their intersection point.Moreover, it must also hold that (1 − ρ 2 V 2 )/ ρ 1 < (1 − q ρ 2 q V 2 )/ q ρ 1 , in order for single-crossing to hold.
As for G 1 , the value of G 1 (x) at x = r and the slope in the range (r, (1 − ρ 2 V 2 )/ ρ 1 ) are lower than those for q G 1 .In order for the two distributions to have equal means, it must then hold that 1 − t / V 2 > 1 − q t / q V 2 .That is, the value of G 1 (x) must be higher for x ∈ [0, r).Hence, again, q G 1 (x) SOSD G 1 (x).
Similarlly to the proof of the previous lemma, the principal prefers it, when both distributions are dominated in the SOSD-sense, hence, he prefers the tiebreaking rule ρ to q ρ.
For part (ii), the decrease of the upper bound on α 2,0 also follows directly from the definition of zone B (α 1,0 , α 2,0 ; ρ 1 , ρ 2 ), where α 2,0 1−r 1−r(1−ρ 1 ) .In order to prove the the statement in terms of bounds on α 1,0 , consider again the system 4.2.3-4.2.5.Apply the implicit function theorem to evaluate dα 1,0 dρ 1 , treating α 1,0 , V 1 , V 2 as endogenous variables, ρ 1 as an exogenous variable, and keeping α 2,0 fixed, and allowing t = 0 for the lower bound and t = 1 for the upper bound.The matrix of derivatives of the system with respect to (α 1,0 , V 1 , V 2 ) T is , with the inverse , and the derivative of the system with respect to ρ 1 the same as before, in (4.2.6).
Overall, we conclude that χ 0 and, thus, C Equilibrium in B under ρ 1 = 0 The mapping established in this paper is not well-behaved for ρ 1 = 0.However, check that the following strategy profile constitutes an equilibrium under ρ 1 = 0 and a valuation w λ i .Every equilibrium of the Information Design game includes the pair of CDFs of posterior beliefs, G * 1 , G * 2 , such that one CDF, G * i , is the maximizer in the Information Design problem (ID) stated above, taking the other CDF, G * k , and also the tie-breaking rule, as given.To every pair (G * 1 , G * 2 ) corresponds a pair of Lagrange Multipliers, (λ * 1 , λ * 2 ).Besides, every such pair constitutes equilibrium CDFs of bids in the all-pay auction with a reserve price r, a bid cap of 1games, G * i maximizes the expected payoff given the opponent's G * k .

Figure 3 :
Figure 3: Equilibrium in zones B, B

Figure 6 :
Figure 6: Change of distributions in response to ρ 1 increase in B

Figure 8 :
Figure 8: Shift of C in response to ρ 1 increase
t ∈ {0, 1}.Hence, we have proven the part (ii) of Lemma 3. The part (i) follows, because the upper bound of B on α 1,0 coincides with the lower bound of D on α 1,0 ; and upper bound of B on α 2,0 coincides with the upper bound of B on α 1,0 .