# Cursed yet Satisfied Agents

In real life auctions, a widely observed phenomenon is the winner's curse – the winner's high bid implies that the winner often over-estimates the value of the good for sale, resulting in an incurred negative utility. The seminal work of Eyster and Rabin [Econometrica'05] introduced a behavioral model aimed to explain this observed anomaly. We term agents who display this bias "cursed agents". We adopt their model in the interdependent value setting, and aim to devise mechanisms that prevent the cursed agents from obtaining negative utility. We design mechanisms that are cursed ex-post IC, that is, incentivize agents to bid their true signal even though they are cursed, while ensuring that the outcome is individually rational – the price the agents pay is no more than the agents' true value. Since the agents might over-estimate the good's value, such mechanisms might require the seller to make positive transfers to the agents to prevent agents from over-paying. For revenue maximization, we give the optimal deterministic and anonymous mechanism. For welfare maximization, we require ex-post budget balance (EPBB), as positive transfers might lead to negative revenue. We propose a masking operation that takes any deterministic mechanism, and imposes that the seller would not make positive transfers, enforcing EPBB. We show that in typical settings, EPBB implies that the mechanism cannot make any positive transfers, implying that applying the masking operation on the fully efficient mechanism results in a socially optimal EPBB mechanism. This further implies that if the valuation function is the maximum of agents' signals, the optimal EPBB mechanism obtains zero welfare. In contrast, we show that for sum-concave valuations, which include weighted-sum valuations and l_p-norms, the welfare optimal EPBB mechanism obtains half of the optimal welfare as the number of agents grows large.

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## 1 Introduction

Consider the following hypothetical game—Alice and Bob each have a wallet, with an amount of money that is known to them, but not to the other player. They each know that the money in the other wallet is distributed uniformly in the range between and , independently of the amount of money in their own wallet. The auctioneer confiscates the two wallets, and runs a second price auction on the two wallets (the highest bidder wins the two wallets, and pays the bid of the other bidder). Say Alice has in her wallet, how should she bid? A naïve strategy Alice could take is to calculate the expected amount of money in Bob’s wallet, , and add it to her amount, resulting in a bid of . However, such a bidding strategy ignores the fact that if Bob invokes the same strategy, then conditioned on Alice winning the two wallets, she has more money in her wallet than Bob, implying that Bob’s amount is distributed uniformly between and . Thus, if both agents invoke the naïve strategy, Alice’s utility conditioned on winning is the expected sum in the two wallets, minus Bob’s expected bid conditioned on him losing , implying an expected negative utility of .

Of course, a rational agent should not incur a negative utility when playing a game. klemperer1998auctions introduced the game presented above, and named it “the wallet game”. This is an interdependent value setting (IDV) milgrom1982theory, where each agent has private information, termed signal (the amount of money in their own wallet), and a public valuation function that takes into account different bidders’ private information (in this case, the sum of signals). klemperer1998auctions analyzed the symmetric equilibrium of rational agents in the wallet game (and introduced several asymmetric equilibria). However, in practice, the observed behavior of agents in the wallet game much resembles the naïve strategy rather than an equilibrium that rational agents end up with avery1997second. This phenomenon was first observed by capen1971competitive, three petroleum engineers who observed that oil companies experienced unexpectedly low rates of returns in oil-lease auctions since these companies “ignored the informational consequences of winning”.

This behavioral bias, where the winner fails to account for the implications of outbidding other agents is commonly referred to as the winner’s curse, and was consistently observed across many scenarios such as selling mineral rights (capen1971competitive; lohrenz1983bonus), book publication rights (dessauer1982book), baseball’s free agency market (cassing1980implications; blecherman1996there), and many others (for more information about empirical evidence for the winner’s curse, see Chapter 1 in kagel2009common

). As standard game theory cannot account for the observed behavior,

eyster2005cursed introduced a behavioral model that formalizes this discrepancy. They termed this model as “cursed equilibrium”.

In their model, agents correctly predict other agents’ strategies, but fail to estimate that other agents’ actions are correlated with their actual signals (or information), similar to the naïve strategy presented above. The extent of this degree of misestimation of the correlation of actions and signals is captured by a parameter , where the perceived utility of agents is times the expected utility of the agents if the actions and signals of other agents are uncorrelated plus times the actual, correct expected utility, correlating the signals with the actions (see Section 3.1 for a formal definition). Having a single parameter to explain the behavioral model of the agents proved to be a very tractable modeling, as this parameter can be easily fitted using real world data to estimate , and better predict players behaviors (eyster2005cursed).

Existing literature analyzes fully rational players’ behavior as they ‘shade’ their bid to account for the potential over-estimation of the value (klemperer1998auctions; milgrom1982theory; wilson1969communications; nobel2021considerations), studies biased agents behavior when deploying known mechanisms, where the equilibrium often times implies a negative utility for the bidders (and higher revenue for the seller) (eyster2005cursed; kagel1986winner; avery1997second; holt1994loser), or exploits the biased behavior of the agents to achieve higher revenue (bergemann2020countering). Our work does not assume agents are fully rational, and views cases where agents experience actual negative utility as undesirable; thus, tries to avoid such scenarios. Indeed, in many real-life scenarios, such as leasing spectrum bands, violating ex-post IR can be detrimental to society at large. Perhaps a mobile company overbids on spectrum, winds up bankrupt, and then the public cannot enjoy any cellular service associated with that spectrum lease (zheng2001high). Another adverse effect that might occur is that companies experiencing revenue loss might feel reluctant to join future auctions (hendricks1988empirical).

In this paper, we design mechanisms that are incentive compatible (IC) for cursed agents—agents maximize their biased utility by reporting their true private information, thus generate a predictable behavior. In order to avoid the winner’s curse, the mechanisms we introduce are ex-post individually rational, meaning an agent will never pay more than their true value. We study the quintessential objectives of revenue and welfare maximization in auction settings with interdependent values.

### 1.1 Our Results

We focus on deterministic and anonymous mechanisms, as they are optimal for interdependent values of fully rational agents (ausubel1999generalized; maskin1992auctions; RoughgardenT16). We extend the cursed-equilibrium model of eyster2005cursed to support a strong truthfulness notion of Cursed ex-post incentive-compatible (C-EPIC), the equivalent of ex-post IC in the case of fully rational agents, which is the strongest incentive notion possible for this setting111The ex-post IC notion is stronger than Bayesian IC and weaker than Dominant strategy IC. In interdependent settings it is impossible to design dominant strategy IC mechanisms while obtaining good performance guarantees. (see Section 3.1). For interdependent values, a deterministic C-EPIC mechanism corresponds to a threshold allocation rule, which takes as input other bidders’ reports, and returns the minimum bid from which an agent starts winning.

After establishing the incentive notion we are studying in this paper, we turn to take a closer look at the implications of ensuring the mechanisms satisfy ex-post individual rationality (Section 5). Our solution concept gives rise to an analogue of the payment identity of EPIC mechanisms (Theorem 1), but as opposed to the fully rational setting, we might need to set the constant term in the payment identity ( in Equation (6)) to be smaller than zero, as the mechanism might make positive transfers to compensate for the over-estimation of values due to the winner’s curse. For a fixed deterministic mechanism, we show how to optimally set this compensation term in a way that maximizes the revenue for the allocation rule, while keeping EPIR. This has the following implication—fixing the allocation rule, and letting grow decreases the revenue. An interesting conclusion is that the revenue and welfare of the revenue-optimal and welfare-optimal mechanisms that are C-EPIC-IR and EPIR (and ex-post budget-balance for welfare) decreases as increase (see Propositions 3 and 4). This is in stark contrast to the case where the mechanism does not require EPIR, where cursed-agents are shown to generate more revenue in second price auctions, since the mechanisms takes advantage of the possibility of agents paying more than their value (eyster2005cursed).

Building upon our understanding of combining individual-rationality constraints with incentive-compatibility for agents who suffer from the winner’s curse, we turn to revenue maximization (see Section 6). We show that designing a revenue optimal C-EPIC-IR and EPIR deterministic mechanism decomposes into a separate problem for each agent and other agents’ signals . That is, the mechanism designer’s task reduces to find an optimal threshold for winning the auction for agent for every set of signals other agents might declare. We show how to optimally set such a threshold, resulting in a revenue optimal mechanism (see Theorem 3). We discuss interesting similarities and differences of the optimal threshold rule from the case where the agents are fully rational, and the case where the seller does not require the mechanism to keep EPIR constraints at the end of Section 6.

Social welfare maximization is a more nuanced task, as just aiming for C-EPIC-IR and EPIR might result in the auctioneer losing a huge amount of money when maximizing, due to the positive transfers the auctioneer needs to make in order to obtain EPIR (see Proposition 7.1). Therefore, when maximizing welfare, we add a requirement of ex-post budget-balance (EPBB); that is, the seller never has negative revenue. A trivial way to ensure EPBB is to set the threshold rule such that it never sells whenever selling implies the seller will need to make positive transfers. We present a masking operation that does exactly this. Given a threshold rule, the masking of the threshold rule allocates in the maximal subset of cases where the original threshold rule made an allocation and the allocating induces no positive transfer from the seller. One might wonder whether one can design a mechanism that is EPBB, while still selling at scenarios that require the mechanism to make positive transfers to some bidders, increasing the expected welfare of the mechanism. We answer this question in a negative way—we show that a mechanism is EPBB if and only if the mechanism never makes positive transfers to any bidder (see Theorem 4). The implication of this theorem is that the welfare-optimal EPBB is a masking of the socially optimal mechanism that is not EPBB (see Proposition 7). We show that for some valuation functions, as the maximum of agents’ signals, this implies the mechanism can never sell the item, resulting in zero welfare. On the positive side, we demonstrate that for Concave-Sum valuations, a family of valuation functions that includes weighted-sum valuations222Weighted-sum valuations take the form for . Note that the valuations in the wallet game example are a special case of weighted-sum valuations. and -norms of signals for a finite , the welfare-optimal EPBB mechanism gets at least half of the fully efficient allocation as the number of agents grows large (see Theorem 5).

## 2 Related work

Our work investigates the problem of auction design for the agents who suffer from the winner’s curse. As a behavioral anomaly (thaler1992winner), the winner’s curse has been documented and analyzed by a large literature via both field studies and lab experiments. Field evidence of the presence of the winner’s curse has been discovered in auction practices across a wide array of industries, which range from the book industry (mcafee1987auctions) and the market of baseball players (cassing1980implications) to the offshore oil-drilling leases (capen1971competitive; hendricks1988empirical; hendricks1987information; porter1995role). Also, a large amount of lab experimental results (bazerman1983won; kagel1986winner; avery1997second; charness2009origin; ivanov2010can; bernheim2019handbook) show that the winner’s curse occurs under various conditions, which differ in multiple dimensions, such as auction format (e.g., first-price, second-price, Dutch auction and English auction), number of participants, valuation functions and signal structures. kagel2002bidding provided a comprehensive review for such experimental studies. Furthermore, most of these studies, such as observational ones (hendricks1988empirical; hendricks1987information) and lab experimental ones (bazerman1983won; kagel1986winner; avery1997second; charness2009origin), demonstrate that the bidders who suffered from the winner’s curse not only experienced a reduce in the profit than anticipated, but could also be worse off upon winning, i.e., receive a negative net profit.

As discussed in the introduction, we adopt the cursed equilibrium model introduced by eyster2005cursed to model the bidding behaviors of agents who suffer the winner’s curse. This model suggests that agents fail to fully appreciate the contingency between other bidders’ bids and the auction item’s value. This cause is supported by several experimental findings (bazerman1983won; kagel1986winner; avery1997second; charness2009origin), and this model has been applied, generalized or analogized to different applications, including analyzing market equilibrium (eyster2019financial; eyster2011correlation), designing financial assets (kondor2015cursed; ellis2017correlation), unifying theoretical behavioral models (miettinen2009partially).

In addition, there is a large literature, including wilson1969communications; milgrom1982theory; klemperer1998auctions; bulow2002prices; RoughgardenT16, studying the interdependent valuation auction, the type of auctions we consider in this paper. Different from our work which designs mechanism for agents who suffer the winner’s curse, these papers consider the mechanism design for fully rational agents who play (Bayesian) Nash equilibrium strategies. milgrom1982theory introduced the interdependent value model and analyzed the revenue of different auction formats when agents have correlated but private value over the item. Their results imply that fully rational agents who implicitly try to avoid the winner’s curse bid more conservatively when there is less information (as in a second price auction) revealed in the auction than more information revealed (as in English auction). bulow2002prices; klemperer1998auctions showed the anomalies in certain interdependent valuation auctions that the item price may increase in supply and decrease in the number of bidders and that the item price is sensitive to even a small asymmetry of bidders. They interpreted the anomalies in terms of fully rational bidders taking the winner’s curse into consideration. RoughgardenT16 developed tools to build ex-post incentive compatible mechanisms for general interdependent valuation auctions with fully rational agents. We use their tools to build our mechanisms.

In contrast to these works, bergemann2020countering studied the auction design problem for agents suffering the winner’s curse. Their work is the most similar to ours, but with several differences: they aim at achieving interim incentive guarantees, while we aim at achieving stronger ex-post incentive guarantees; they consider agents who are fully cursed, while we consider agents who can be partially cursed; they consider a single valuation function for the item, defined as the maximum of all agents’ private signals, while we consider a more general family of valuation functions.

Finally, the design of mechanisms when considering agents who act according to a behavioral bias, and not their objective utility, have recently gained traction. Recent examples of this line of research are finding a market equilibrium for agents who suffer from the endowment effect (EzraFF20; BabaioffDO18) and designing revenue-maximizing auctions for agents who are uncertainty-averse (0001GMP18; LiuMP19), among others.

## 3 Model

We consider a seller that sells a single indivisible item to a set of agents with interdependent valuations (the IDV model). Each agent has a signal

as private information. The agents’ signals are drawn from a joint distribution

with density over the support , where . We use to denote the agents’ signal profile and use to denote the signal profile of all agents except agent . We impose the standard assumptions that is continuous and nowhere zero on the signal space. Each agent also has a publicly-known valuation function , which represents the value received by agent upon winning the auction as a function of all bidders’ signals. We adopt the following standard assumptions in the valuation function :

• Non-negative and normalized, i.e., and .

• Continuously differentiable.

• Monotone-non-decreasing in all signals and monotone-increasing in agent ’s signal .

We consider the case of symmetric agents (as in eyster2005cursed; milgrom1982theory; RoughgardenT16). We assume:

1. (Signal symmetry) , where for some , and for any signal profile and its arbitrary permutation .

2. (Valuation symmetry) For any , as long as and is a permutation of .

As standard in the interdependent literature, when devising the welfare-optimal mechanism, we assume that the valuations satisfy the single-crossing condition, presented here in the context of symmetric valuation functions.

###### Definition 1 (Single-Crossing for symmetric valuation functions).

Symmetric valuation functions satisfy the single crossing condition if for any signal profile and agents , if and only if .

We use this common assumption for the results regarding the characterization and analysis of the welfare-optimal mechanism. (Theorem 5).

By the revelation principle, we consider without loss of generality direct mechanisms, in which agents directly report their private signals and then the auctioneer determines the auction outcome according to a pre-announced mechanism . Here, is the allocation rule, specifying agent

’s winning probability, and

is the payment rule, specifying agent ’s payment.

We study deterministic and anonymous mechanisms, as these are optimal in our setting for non-cursed agents (RoughgardenT16; ausubel1999generalized; maskin1992auctions). A mechanism is deterministic if for any and . A mechanism is anonymous if for any and any permutation of , whenever .

We make two technical assumptions about allocation rule for simplicity of exposition: First, we do not allocate the item to an agent who reports a zero signal.

###### Assumption 1.

for every and .

Second, we do not allocate when there is a tie in the highest reported signals.

###### Assumption 2.

for every whenever .

Since is continuous, the events in both assumptions have zero probability measure, and therefore, can be ignored without affecting the expected social welfare or revenue of the mechanism. Moreover, these assumptions are without loss for the mechanisms we consider as implied by Lemma 2 in Section 4.2.

We use to denote the reported signal profile (bid profile) of agents. Agents have quasilinear utilities—the utility of each agent given private signal profile and bid profile under mechanism is

 ui(b,s)=xi(b)vi(s)−pi(b).

### 3.1 The Winner’s Curse—A Behavioral Model

We adopt the widely studied behavioral model, namely the cursed equilibrium model, introduced by eyster2005cursed to explain the occurrence of the winner’s curse. In this model, agents fail to incorporate the contingency between the other bidders’ actions and their signals, which determine the value of the auctioned item, but succeed in reasoning other parts of the game. To illustrate, let denote a bidding strategy profile of agents and denote the probability density of bids and signals under strategy profile , e.g., represents the probability density of other bidders bidding when having signals . Given the strategy profile , a fully rational agent with signal estimates the probability density of other agents receiving and bidding as

 fσ(b−i,s−i|si)=f(s−i|si)fσ(b−i|s−i).

Consequently, suppose the other agents follow strategy , such an agent estimates her expected utility when having signal and bidding as follows:

 EUi(bi,si;σ−i)=

In contrast, an agent who fully neglects the contingency between other agents’ actions and their signals estimates, as the naïve agent in the wallet game example, the counterpart probability density as if and are independent conditioned on her knowledge :

 ~fσ(b−i,s−i|si)=f(s−i|si)fσ(b−i|si),

where 333eyster2005cursed suggested that the agents succeed in reasoning or perceiving all other parts of the game, except the contingency between other agents’ signals and actions. Therefore, agents get the correct , a key assumption made in Eyster and Rabin’s behavioral model. eyster2005cursed further introduce a cursedness parameter to model the case where an agent partially neglects this contingency such that she considers the counterpart probability density as

 fχσ(b−i,s−i|si)=(1−χ)fσ(b−i,s−i|si)+χ~fσ(b−i,s−i|si).

An agent with is a fully rational agent, and as we will see later, an agent with is possible to experience the winner’s curse. We refer to such an agent with as a cursed agent for short, and to an agent with as a fully cursed agent. By analogy with a fully rational agent estimating her expected utility, an agent with parameter (falsely) estimates her expected utility given as:

 EUχi(bi,si;σ−i)=∫s−i∈Sn−1∫b−i∈Sn−1fσχ(b−i,s−i|si)(xi(b)vi(s)−pi(b))db−ids−i (1)

To explain and predict the winner’s curse phenomenon, eyster2005cursed suggested that agents generally play the equilibrium strategy with respect to this misperceived utility for some parameter , instead of . They referred to this equilibrium as the -cursed equilibrium. Formally, a strategy profile forms a cursed equilibrium if it holds that for every agent ,

 σ(bi|si)>0⟺bi∈argmaxEUχi(bi,si;σ).

In the above definition, gives the definition of the classic Bayesian Nash equilibrium (BNE). The Naïve strategy of the aforementioned wallet game is an example of a cursed equilibrium of fully cursed agents (). We refer the reader to Appendix A for an illustrative example of how a -cursed equilibrium leads to a winner’s curse in the wallet game.

Next, we illustrate how -cursed equilibrium relates to the winner’s curse. Note that Eq. (1) can be rewritten444This rewriting result is given by eyster2005cursed. We present a derivation in the Appendix for completeness. as follows:

 EUχi(bi,si;σ−i)=∫s−i∈Sn−1∫b−i∈Sn−1f(s−i|si)⋅fσ(b−i|s−i)(xi(b)vχi(s)−pi(b))db−ids−i, (2)

where

 vχi(s):=(1−χ)vi(s)+χE~s−i[vi(~s−i,si)]. (3)

This rewriting shows that the utility optimized by an agent with valuation function in the -cursed equilibrium is the same as the expected utility optimized by a fully rational agent with valuation function in a BNE. Thus, we have the following proposition from eyster2005cursed.

###### Proposition 1 (eyster2005cursed).

In the IDV setting, the -cursed equilibrium strategy profile of agents with valuation function is the same to the BNE strategy profile of agents with a modified valuation function .

We name the expression as the cursed valuation function of . It reflects the hypothetical value of the item to the cursed agent. It contains two part—the part reflects the part of the item’s value which the agent perceives through successful contingent thinking and the part reflects the part of the item’s value which the agent perceives when she fully ignore the contingency between other bidders’ signals and bids. Therefore, a winner faces the winner’s curse whenever ; i.e., bidder might get an item of which the value is less than anticipated. Moreover, might suffer a negative utility when the payment, which can be as high as for a fully cursed agent, turns out larger than . We refer to the conceptual utilities built upon the cursed valuation functions as cursed utilities.

eyster2005cursed showed with empirical wallet game data that

has a 95% confidence interval of [0.59, 0.67] with 0.63 the optimal fit. Any

predicts agents’ bids better than the BNE strategy.

We make the following key assumption:

###### Assumption 3 (Seller knows χ).

We assume the seller knows the value of , that is, the seller knows the extent of which the agents exhibit the cursedness bias.

The above assumption can be justified by the following: (i) empirical studies discussed above, showing one can accurately estimate the value of ; (ii) moreover, the seller can observe the practical behavior of the agents, and their profit, in order to adjust the value of , and update the devised auction, if the estimated value of seems to be inaccurate. If the assumption does not hold, it is still worthy studying the problem under this assumption for following reasons. First, we show that misestimating the value of by still leads to an approximate C-EPIC-IR mechanism (Proposition 2), and thus using with a small estimation error still preserves some degree of incentive compatibility. Second, devising mechanisms when assuming knowing the value of leads to many interesting theoretical findings, which have implications on designing mechanisms for agents who suffer from winner’s curse. An example of one such implication is that there exists a tension between ensuring that the agents would not experience negative profit due to their inability to reason about their utility and the revenue of the mechanism. In order words, to ensure non-negative utility for agents who may suffer from the winner’s curse, the mechanism has to sacrifice some portion of the revenue.

### 3.2 Incentive Properties for Cursed Agents and Other Desirable Properties

Bearing above behavioral implications of agents with parameter in mind, a natural generalization of the interim IC concept from fully rational agents to agents with parameter is the following.

###### Definition 2.

A mechanism is interim incentive compatible for agents with parameter , if for all , and for the truth-telling strategy , it holds that

 EUχ(si,si;σ∗−i)≥EUχ(bi,si;σ∗−i).

In other words, a mechanism is interim incentive compatible for agents with parameter if truthful-reporting is a -cursed equilibrium. For fully rational agents (=0), the above definition coincides with the standard interim IC definition (RoughgardenT16), where truthful-reporting forms a BNE (that is, a -cursed equilibrium).

We further extend this idea to obtain a stronger IC notion. We consider a cursed agent’s expected utility when having signal , while the bid profile is . A fully rational agent will correctly estimate the degree to which other agents’ signals are contingent on their bids , setting this probability as , while a fully cursed agent will think the true type is independent of agents’ bids, estimating this probability as . Therefore, an agent with parameter will assess her expected utility given her signal , bid and others bidding given strategy as:

 EUχi(b,si;σ−i) =

Note that we have the following relationship between and :555We present the derivation of Equation (4) in Appendix C.

 EUχi(bi,si;σ−i)=∫b−i∈Sn−1fσ(b−i|si)EUχi(b,si;σ−i)db−i. (4)

Therefore, we can naturally define the ex-post incentive properties for agents with parameter as follows.

###### Definition 3 (Cursed ex-post incentive compatibility and individually rationality (C-Epic-Ir)).

Given a cursedness parameter , a mechanism is cursed ex-post incentive compatible (C-EPIC) if for every , and , and truthfully-reporting strategy ,

 EUχi(b=s,si;σ∗−i) ≥ EUχi((b−i=s−i,bi),si;σ∗−i),∀bi.

A mechanism is cursed ex-post individually rational (C-EPIR) if for every

 EUχi(b=s,si)≥0.

A mechanism that is both C-EPIC and C-EPIR is denoted by C-EPIC-IR.

Obviously, C-EPIC implies the interim IC for agents with parameter .

Lemma 1 introduces an equivalent definition of C-EPIC-IR, which simplifies the analysis of whether a mechanism satisfies C-EPIC-IR or not.

###### Lemma 1.

A mechanism is C-EPIC if and only if for every , and ,

 xi(s)vχi(s)−pi(s)≥xi(bi,s−i)vχi(s)−pi(bi,s−i)∀bi.

A mechanism is C-EPIR if and only if for every

 xi(s)vχi(s)−pi(s)≥0.
###### Proof.

To see this lemma holds, we only need to plug the following expression of the expected utility of bidders into Definition 3:

 EUχi((b−i=s−i,bi),si;σ∗−i) (5) = ∫~s−i∈Sn−1((1−χ)fσ(~s−i|s−i,si)ui((bi,s−i),(si,~si)) +χf(~s−i|si)ui((bi,s−i),(si,~s−i)))d~s−i = (1−χ)ui((bi,s−i),s) +χ∫~s−if(~s−i|si)(vi(~s−i,si)xi(bi,s−i)−pi(bi,s−i))d~s−i = xi(bi,s−i)((1−χ)vi(s)+χ∫~s−if(~s−i|si)vi(si,~s−i)d~s−i)−pi(bi,s−i) = xi(bi,s−i)((1−χ)vi(s)+χE~s−i∼F|si[vi(~s−i,si)])−pi(bi,s−i) = xi(bi,s−i)vχi(s)−pi(bi,s−i),

where in the last equation is the cursed valuation function of the item, as defined in Eq (3). ∎

Setting in the definition of C-EPIC gives us the definition of ex-post IC (EPIC), where bidders truthfully reporting their signals forms an ex-post Nash equilibrium w.r.t. their true ex-post utilities. It is the strongest incentive guarantee one can hope for in the IDV setting. Similarly, C-EPIC is also the strongest incentive notion we can hope for with cursed agents in the IDV setting. Furthermore, Proposition 2 shows that C-EPIC is robust to small estimation errors of the parameter.

###### Proposition 2.

Let mechanism be C-EPIC under cursedness parameter , and let agent ’s be a -cursed agent, where . The truthful-reporting strategy forms an approximate ex-post Nash equilibrium for agent with parameter in the sense that

 EUχii(b=s,si;σ∗−i)≥EUχii((b−i=s−i,bi),si;σ∗−i)−ϵi⋅vi(¯s,...,¯s)∀i,s,bi.
##### Ex-post IR (Epir)

Setting in the definition of C-EPIR gives us the standard definition of EPIR, which ensures that no bidder will get a true negative ex-post utility at the truthful-reporting equilibrium. A mechanism that is C-EPIC-IR has the outcome that every agent bidding their true signal is a cursed equilibrium (or an ex-post equilibrium in terms of their cursed utilities) with each agent obtaining a non-negative utility based on their cursed valuation functions. However, although the agents think their utility will be non-negative for any possible realization of signals according to their belief, they might end up paying more than their value for the item leading to a negative utility, because their belief is inaccurate. Therefore, in addition to requiring C-EPIC-IR, we further consider designing mechanisms that are EPIR. Such mechanisms guarantee the agent will not experience actual negative utility upon receiving an item, therefore, such an agent would not regret participating in the auction in hindsight.

##### Ex-post budget balance (Epbb)

In order to achieve the EPIR property, the mechanism might need to make positive transfers since the agents over-estimate their value for the item sold. In order to ensure the seller does not end up with negative revenue, we may also want to require that the mechanisms will satisfy the ex-post budget balance constraint.

###### Definition 4 (Ex-post budget-balance).

A mechanism is ex-post budget-balanced (EPBB) if for every signal profile ,

A more relaxed requirement is Ex-ante budget-balance, where the mechanism does not lose money in expectation.

When devising a mechanism that satisfies C-EPIC-IR, there is a natural tension between EPIR and budget-balance. The socially optimal mechanism might have negative revenue when satisfying EPIR (see Section 7.1)). Moreover, while typical mechanisms usually have more revenue with cursed agents (without imposing EPIR(eyster2005cursed), when requiring the mechanism to satisfy EPIR, the revenue only decreases (see Proposition 3).

## 4 Preliminaries

### 4.1 C-Epic-Ir Mechanisms and Virtual Valuations

RoughgardenT16 introduced the analogue of Myerson’s payment identity for the IDV model. We observe that a simple adaptation of their results characterizes the space of C-EPIC-IR mechanisms. We omit the derivation, as it is identical to the one in RoughgardenT16 for the case of non-cursed agents.

###### Theorem 1.

A mechanism is C-EPIC-IR if and only if for every , , the allocation rule is monotone non-decreasing in the signal , and the following payment identity and payment inequality hold:

 pi(s)=xi(s)vχi(s)−∫vχi(s)vχi(0,s−i)xi((vχi)−1(t|s−i),s−i)dt−(xi(0,s−i)vχi(0,s−i)−pi(0,s−i)) (6)
 pi(0,s−i)≤xi(0,s−i)vχi(0,s−i) (7)

In the setting where valuations are not cursed, setting maximizes the seller’s revenue, and makes sure that the seller never has to pay the buyers participating in the auction, therefore ensures that the mechanism is budget-balanced. However, for cursed agents, even though , it might as well be the case that resulting in negative utility, and breaching the EPIR property. Therefore, fixing a mechanism, one might want to set to be strictly smaller than zero for some values of , which means the mechanism might pay agents for participating. Thus, in designing a mechanism to guarantee EPIR, one must take care in order not to violate budget balance.

RoughgardenT16 define the a virtual valuation for interdependent values. Given , they define a function

 φi(si|s−i)=vi(s)−vi′(si,s−i)1−F(si | s−i)f(si | s−i),

and show that similarly to the private value setting, revenue maximization reduces to virtual welfare maximization. The definition of virtual valuations and formulating revenue maximization as virtual welfare maximization naturally extends to the case of cursed bidders.

###### Definition 5 (Cursed virtual value).

The cursed virtual valuation of agent conditioned on is defined as

 φχi(si | s−i)=vχi(s)−vχi′(si,s−i)1−F(si | s−i)f(si | s−i).

The next theorem follows the exact same derivation as the one in RoughgardenT16 for non-cursed agents.

###### Theorem 2 (Follows from RoughgardenT16).

For every interdependent values setting, the expected revenue of a C-EPIC-IR mechanism equals its expected conditional cursed virtual surplus, up to an additive factor:

 Es[∑ipi(s)]=Es[∑ixi(s)φχ(si|s−i)]−∑iEs−i[xi(0,s−i)vχi(0,s−i)−pi(0,s−i)]

### 4.2 Deterministic C-Epic-Ir Mechanisms

In this paper we focus on deterministic mechanisms, as deterministic mechanisms are optimal for our setting whenever bidders are not cursed (RoughgardenT16). The following is a direct corollary of the monotonicity of C-EPIC-IR mechanisms.

###### Corollary 1.

Any deterministic C-EPIC-IR mechanism is a threshold mechanism. i.e., for every , there exists a function such that

We will refer to as the critical bid for agent . The following lemma restricts the set of allocation rules we inspect.

###### Lemma 2.

For every deterministic, anonymous C-EPIC mechanism and for every , if the items is allocated, it is allocated to a bidder in .

###### Proof.

Assume the item is given to an agent such that there exists for which is strictly bigger than . By anonymity, there exists some such that if we switch and ’s signal, wins the item. Since wins at , also wins at by monotonicity of C-EPIC mechanisms. Since wins at , by monotonicity, also wins at , a contradiction. ∎

The above lemma implies that when dealing with such mechanisms, assuming Assumptions 1 and 2 are without loss. A zero signal cannot win unless it is the highest signal. Moreover, one can take any mechanism that violates this assumption, and set for every where there is a tie in the highest bid. By the above lemma, such mechanism remains monotone non-decreasing in a bidder’s own bid, and therefore C-EPIC.

#### 4.2.1 Generalized Vickrey Action

We present the welfare optimal mechanism for the IDV setting with non-cursed agents. This mechanism is fully efficient and EPIC-IR if the valuations satisfy the single-crossing condition (see Definition 1. This mechanism is the basis of our welfare-optimal EPBB mechanism for cursed valuations, named masked GVA.

###### Definition 6 (Generalized Vickrey Auction (GVA) (maskin1992auctions; ausubel1999generalized)).

The generalized Vickrey auction elicits bidders to bid their private signals. The winner is a bidder in , and the payment is , where . The GVA auction is EPIC-IR and maximizes welfare when valuations satisfy the single-crossing condition.666A single-crossing condition basically states that a bidder who has the highest at a signal profile continues to have the highest value as increases. For symmetric settings, this mechanisms allocates to the bidder with the highest signal, and is the second highest signal.

## 5 Implications of Ex-post IR

In this section we discuss implications of imposing EPIR on the mechanism. We first show that in order to achieve our incentive properties, it suffices to design an ex-post IR mechanism, and cursed ex-post IR will follow.

###### Lemma 3.

For every interdependent value setting, C-EPIC and EPIR implies C-EPIR.

###### Proof.

For a mechanism to be EPIR, we need that the actual value an agent gets from an allocation is higher then the price she pays. That is, for every . Using equation (6) for C-EPIC mechanisms, and rearranging, we get that for every and every

 pi(0,s−i)≤xi(s)(vi(s)−vχi(s))+∫vχi(s)vχi(0,s−i)xi((vχi)−1(t|s−i),s−i)dt+xi(0,s−i)vχi(0,s−i). (8)

Specifically, fixing and setting , we get

 pi(0,s−i) ≤ = xi(0,s−i)vχi(0,s−i),

where the inequality used Assumption 1, that agents aren’t allocated at their lowest signal (). This coincides with Equation (7), implying C-EPIR. ∎

According to Corollary 1, every deterministic allocation rule is equivalent to a set of threshold functions . As noted before, the only freedom one have in setting payments of C-EPIC mechanisms is by setting the term . We show that when maximizing revenue subject to C-EPIC and EPIR constraints, there is a single optimal way to set .

###### Lemma 4.

Fixing threshold functions of a deterministic anonymous mechanism, the revenue optimal C-EPIC-IR and EPIR mechanism sets

 pi(0,s−i)=min{0,vi(ti(s−i),s−i)−vχi(ti(s−i),s−i)}.

(and therefore, the payment is uniquely defined using Equation (6).)

###### Proof.

Since by Assumption 1, we have to have , otherwise an agent pays without getting allocated, which leads to negative utility. Combining with Equation (8) for EPIR (which also implies C-EPIR), we get

 pi(0,s−i) = min{0,minsi{xi(s)(vi(s)−vχi(s))+∫vχi(s)vχi(0,s−i)xi((vχi)−1(t|s−i),s−i)dt}} = min{0,minsi>ti(s−i){vi(s)−vχi(s)+∫vχi(s)vχi(ti(s−i),s−i)1dt}} = min{0,minsi>ti(s−i){vi(s)−vχi(s)+(vχi(s)−vχi(ti(s−i),s−i))}} = min{0,minsi>ti(s−i){vi(s)−vχi(ti(s−i),s−i)}} = min{0,vi(ti(s−i),s−i)−vχi(ti(s−i),s−i)}

where the second equality follows from the fact that only if .∎

We get that when agents experience the winner’s curse at the “critical value” (meaning their real value is smaller than their perceived value), they pay their real value at the critical bid, while if they do not experience the winner’s curse, they pay their cursed value. We get the following corollary.

###### Corollary 2.

Fixing an anonymous, deterministic, C-EPIC-IR, EPIR mechanism with threshold function , the revenue optimal way to set gives the following payment function:

 pi(s)=⎧⎪⎨⎪⎩pi(0,s−i)si≤ti(s−i)vi(ti(s−i),s−i)si>ti(s−i) and vi(ti(s−i),s−i)−vχi(ti(s−i),s−i)≤0vχi(ti(s−i),s−i)si>ti(s−i) and vi(ti(s−i),s−i)−vχi(ti(s−i),s−i)>0.
###### Proof.

If , then does not get the item, and by Equation (6), . If , then gets the item, and according to Equation (6) and Lemma 4,

 pi(s) = vχi(ti(s−i),s−i)+pi(0,s−i) = vχi(ti(s−i),s−i)+min{0,vi(ti(s−i),s−i)−vχi(ti(s−i),s−i)}.

Therefore, if , , and if , then

An interesting implication of this corollary is that as opposed to the case where we do not require EPIR, the welfare and revenue in the case of non-cursed agents (i.e., ) is at least that of the case where agents are cursed (). We show this more generally by showing that the welfare and revenue are monotonically non-increasing in . In order to show this, we first show that for a fixed mechanism, EPIR implies that the revenue decreases as increases.

###### Lemma 5.

Fix a threshold rule . Then for every , every , and every , , where is the optimal payment an agent with cursedness parameter has with signals .

###### Proof.

We prove by the three cases of Corollary 2.

Case 1: . Then, , and . Notice that if , then

 vi(t(s−i),s−i) < vχi(t(s−i),s−i) = ⟺vi(t(s−i),s−i) < Et−i∼F|t(s−i)[vi(t(s−i),t−i)] ⟺vχi(t(s−i),s−i) = ≤ (1−χ′)vi(t(s−i),s−i)+χ′Et−i∼F|t(s−i)[vi(t(s−i),t−i)] = vχ′i(t(s−i),s−i) ⟺pχi(s,t) = (9) ≥ vi(t(s−i),s−i)−vχ′i(t(s−i),s−i)