{'fraction_non_alphanumeric': 0.05569037988738203, 'fraction_numerical': 0.015306527083829012, 'mean_word_length': 4.333389730141274, 'pattern_counts': {'":': 0, '<': 31, '<?xml version=': 0, '>': 35, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
First-Price Auctions with Budget Constraints

Consider a first-price, sealed-bid auction with interdependent valuations and private budget constraints. Private budget constraints introduce subtle strategic tradeoffs with first-order consequences for equilibrium bidding. In a pure-strategy, symmetric equilibrium, agents may adopt discontinuous bidding strategies resulting in a stratification of competition along the budget dimension. In an asymmetric setting, equilibria in “nondecreasing” strategies exist, albeit in a qualified sense. Private budgets introduce significant confounds for the interpretation of bidding data due to their interaction with risk preferences and their countervailing strategic implications.


Introduction
Private budgets arise in many economic situations, including auctions. Once bidders have multidimensional private information involving a value-signal, which informs their preferences, and a budget, which defines their feasible strategy set, even simple bidding games exhibit nontrivial equilibria. Many familiar intuitions require qualification or reassessment.
For concreteness, consider a first-price, sealed-bid auction for one item. Bidders simultaneously place bids, the highest bidder wins, and he pays his own bid. Private budgets affect equilibrium bidding in two opposing ways in this familiar setting. First, budgets dampen bids by introducing a spending limit. Che and Gale (1998) characterize this feature in their seminal analysis of the first-price auction with private budget constraints. 1 Assuming private values, they propose sufficient conditions ensuring equilibrium bidding strategies assume the following canonical form: In (1), s i is bidder i's private value, w i is his private budget, andb(·) is an increasing, continuous function that equals the bid of an "unconstrained" agent who bids less than his budget.
Second, and less intuitively, budgets can amplify certain agents' equilibrium bids. A higher bid may be disproportionally effective at defeating budget-constrained rivals, thus giving a high-budget bidder a new strategic option. Exercising this option can lead to an equilibrium where agents adopt discontinuous bidding strategies, even in a symmetric setting. A stratification of competition between high-and low-budget bidders emerges in equilibrium. Hitherto under-appreciated in the literature, these strategic implications complicate equilibrium bidding and affect the auction's efficiency and revenue potential.
To illustrate our argument, it is helpful to consider an example conveying its central intuition. The following is an extension of Vickrey's (1961) model. 2 Example 1. Consider a first-price, sealed-bid auction for one item. There are two ex ante identical bidders. Each bidder i observes an independent, uniformly-distributed signal s i ∈ [0, 1] regarding his value for the item. Signals are private, but their distribution is common knowledge. If a bidder with value-signal s i wins the auction with the bid b i , his payoff is s i − b i ; otherwise, it is zero. A fair coin flip resolves ties. It is well-known that in this auction's Bayesian-Nash equilibrium both bidders adopt the strategy s i /2. Suppose, additionally, that each bidder has a budget w i , which is independent of his valuesignal. Budgets are private, but their distribution is common knowledge. With probability 1/2, a bidder's budget is 1/4; with probability 1/2, it is 3/4. A budget is a hard constraint on bids and a type-(s i , w i ) bidder cannot bid above w i .
A tempting conjecture is that is a symmetric equilibrium strategy in this enriched setting. In (2), which has the canonical form, a bidder follows the usual equilibrium strategy s i /2 until he exhausts his funds. Though intuitive, (2) cannot be a symmetric equilibrium strategy. If bidder j follows (2), then bidder i has a profitable deviation when s i = 1/2 − ϵ and w i = 3/4. According to (2), he should bid 1/4 − ϵ/2. But, if he bids ϵ more, 1/4 + ϵ/2, his probability of winning jumps from approximately 1/2 to over 3/4. At infinitesimal added cost, this deviation is worth it. 3 Reflecting on the preceding reasoning, we are led to an equilibrium strategy with jump discontinuities. Figure 1 sketches this auction's symmetric equilibrium strategy, wheres = 1/3 ands ′ = 11/27. 4 Clearly, this strategy departs from the canonical form.
3 Similar reasoning applies whenever the bid distribution has a mass point due to a mass point in the budget distribution in the relevant range of bids. When this is the case, the first-price auction cannot have an equilibrium in continuous strategies. 4 We verify that the defined strategy is a symmetric equilibrium in Online Appendix A. We show that the equilibrium's qualitative characteristics are unchanged when the distribution of valuations is H(s) and 2 Related Literature Che and Gale (1998) were the first to study standard auctions with private values and private budget constraints. 6 They investigate equilibria satisfying the canonical form (1). In Section 3.1 we study canonical equilibria in a complementary class of cases, including those with interdependent values. Parreiras (2002, 2003) study the second-price auction with private budgets. We discuss their results in detail in Section 3.4. In a similar setting, Kotowski and Li (2014a,b) analyze all-pay auctions. Ghosh et al. (2018) study wars of attrition with private budgets.
In developing our model, we suppress several embellishments commonly associated with budget constraints. Unlike Zheng (2001), Jaramillo (2004), or Rhodes- Kropf and Viswanathan (2005), we model budgets as hard bounds on bids, rather than as imperfections in bidfinancing ability. We also abstract from the risk of collusion or the formation of bidding consortia due to liquidity limitations (Cho et al., 2002). Burkett (2016) argues that budget constraints solve a principal-agent problem when bidding is delegated to a third party.
Budgets are exogenous in our model, as they would be in the absence of delegation. Interest in the role of budgets in auctions has grown due to their appearance in many Internet-related applications (Ashlagi et al., 2010;Balseiro et al., 2015) and spectrum auctions (Cramton, 1995;Bulow et al., 2017). Noting these applications, several authors build models investigating budget constraints in sequential or multi-unit auctions (Pitchik and Schotter, 1988;Benoît and Krishna, 2001;Pitchik, 2009;Lopomo, 2008, 2009;Ghosh, 2015;Kariv et al., 2018). Our model is not a special case of any of them.
The auction design problem following Myerson (1981) incorporating budget constraints is particularly challenging and has been considered under various guises (Laffont and Robert, 1996;Monteiro and Page, 1998;Gale, 1999, 2000;Maskin, 2000;Malakhov and Vohra, 2008;Pai and Vohra, 2014;Kojima, 2014;Baisa, 2018;Boulatov and Severinov, 2018;Carbajal and Mu'alem, 2018;Richter, 2019). Auction design with budgets has also spurred an interest among computer scientists. 7 Our analysis takes the auction format as given and examines equilibrium bidding. Che and Gale (2006) examine auction revenue when bidders have multidimensional types, which may include private financial constraints.

Symmetric Auctions with Interdependent Values
Consider the following adaptation of  model of a first-price auction. There are two ex ante symmetric bidders. Let s i ∈ [0, 1] be the value-signal of bidder i. It defines his information about the item for sale. Value-signals are independently and identically distributed according to the twice continuously-differentiable cumulative distribution function (c.d.f.) H(s i ). (We relax independence in Remark 3 below.) The associated probability density function (p.d.f.) h(s i ) is strictly positive and bounded. Given s i and s j , v(s i , s j ) is bidder i's valuation for the item. The valuation function is continuously differentiable, strictly increasing in the first argument, nondecreasing in the second, v(0, 0) = 0, and v(1, 1) = v. If bidder i wins the auction with the bid b i , his payoff is v(s i , s j ) − b i ; otherwise, it is 0. Ties are resolved with a uniform randomization.
When budget constraints are absent,  show that there exists a Bayesian-Nash equilibrium where all bidders adopt a common strategy, α(s i ), that solves subject to α(0) = 0. Henceforth, we reserve the notation "α(·)" for this specific strategy. Generalizing the model, let w i ∈ [w, w] be the budget of bidder i. Budgets are independently and identically distributed according to the twice continuously-differentiable c.d.f. G(w i ). We assume that 0 ≤ w < v < w and that the associated p.d.f. g(w i ) is strictly positive and bounded for all w i ∈ [w, w]. For convenience, we define G(w i ) = 0 if w i < w and G(w i ) = 1 if w i > w. A bidder's type (s i , w i ) is private information; otherwise, the environment is common knowledge. In this section we focus on a symmetric Bayesian-Nash equilibrium and we suppress bidder subscripts.  bidder j follows a canonical strategy, the expected payoff of a type-(s, w) bidder i when he

A Canonical Equilibrium
The first term in (4) is the expected payoff from defeating an opponent with a value-signal less than x. The second term is the expected payoff from defeating an opponent with a valuesignal greater than x who has a budget less thanb(x). This term vanishes ifb(x) < w. By computing the first-order condition, dU(b(x)|s, w)/dx x=s = 0, and defining the expressions we can characterizeb(s) at its points of differentiability as follows: Whenb(s) < w, (5) reduces to (3), the differential equation characterizing the equilibrium strategy in the absence of budget constraints. To interpret (5) whenb(s) > w, we expand the equation and rearrange its terms. Omitting the function arguments for readability, . (6a)

Terms [A] and [B]
are the expected marginal benefit of an infinitesimally higher bid. When the higher bid defeats an opponent with a value-signal less than s, the payoff gain is v(s, s) − b(s); otherwise, when it defeats an opponent with a budget less thanb(s), the payoff gain is is the expected marginal cost of a higher bid. Further rearrangement of (6a) gives .
Forb ′ (s) to be positive, as assumed throughout its derivation, both numerator and denominator in (6b) must have the same sign. When the budget distribution is relatively dispersed, and say g(b) ≈ 0, this is not a demanding requirement as the denominator is positive. If the budget distribution is concentrated near some value, and g(b) is relatively large, the denominator of (6b) may approach zero, or be negative. Intuitively, when this occurs the likelihood of defeating a budget-constrained opponent is large and a (high-budget) bidder wishes to increase his bid aggressively to capitalize on this opportunity. In Example 1, the return to a slightly-higher bid spiked at the atom in the budget distribution at 1/4 and lead to a discontinuous jump in a bid. As (6b) reveals, an atom is not necessary to provoke a similar response.
Motivated by the preceding discussion, we introduce a regularity condition on the denominator of (5) ensuring the marginal returns to a slightly higher bid are moderate and well-behaved. As explained below, it is a sufficient condition for a canonical equilibrium to exist when w = 0.
Assumption 1. For each s, the function b → η(s, s) − δ(b, s) crosses zero at most one time.
Remark 2. Assumption 1 is implied by Assumption 5 of Che and Gale (1998) when the latter is applied to the case of independent budgets and value-signals, as in our model. 9 Assumption 1 has two parts. The first says that the denominator of (5) satisfies a singlecrossing condition. Since η(s, s) − δ(w, s) < 0, any crossing must be from above. Thus, the second part is a technical restriction ensuring the crossing is strict.
Theorem 1. Suppose Assumption 1 holds and w = 0. There exists an equilibrium where each bidder adopts a common strategy, β(s, w) = min{b(s), w}. The functionb(·) is increasing, continuous, and for a.e. s,b Generically, the equilibrium is unique in the class of symmetric equilibria in canonical strategies.
We provide a numerical example illustrating an equilibrium in Section 3.3. We devote this subsection's remainder to the proof of Theorem 1.

Construction of a Canonical Equilibrium
To prove Theorem 1, we must confirm the functionb(·) exists with the stated properties. A natural approach is to (try to) solve (7) assumingb(0) = 0. This is premature. With interdependent values, zero may not be an optimal bid for an unconstrained bidder. A small bid may defeat a budget-constrained opponent with favorable information concerning the item's value. This attenuation of the winner's curse can inflate an agent's bid, even if he observes a low value-signal. Moreover, (7) may lack an increasing solution that is defined for all s, as assumed throughout its derivation. 9 Che and Gale (1998) invoke their Assumption 5 to prove their Lemma 1. Fang and Parreiras (2002, footnote 7) point out that Che and Gale's lemma may not be true without additional assumptions in their model. When applied to our model, Che and Gale's assumption is sufficient for the existence of an equilibrium in canonical strategies when w = 0.
Acknowledging the above complications, we identifyb(·) indirectly by focusing on the qualitative behavior of solutions to (7). To do so, we use phase-plane analysis to study the closely-related plane-autonomous systeṁ w, w]. 10 In analyzing this system, we are interested in the graphs of its solutions and not in its dynamics. Graphs of its solutions can be identified with integral curves of (7) since, following Birkhoff and Rota (1978, Chapter 5), we observe that Interpreting the problem as proposed lets us use graphical techniques to discern the behavior of candidate solutions over a large domain (Strogatz, 1994;Perko, 2000). Additionally, our reinterpretation placates the singularities encountered in a direct analysis of (7).
Analysis of system (8) focuses on its nullclines and critical (or fixed) points. The nullclines are These graphs characterize points whereṡ = 0 andḃ = 0, respectively. Their economic interpretation is the following. At b = ψ(s) the denominator of (7) changes sign. When η(s, s) − δ(b, s) > 0, marginally higher bids are relatively effective (conditional on s) at defeating a budget-constrained opponent. Otherwise, this dimension's marginal contribution to a bidder's payoff is more subdued. By definition, ν(s) = v(s, s). Thus, ν(s) is a type-s bidder's valuation conditional on his opponent observing the same value-signal. If a type-s agent bids more than ν(s), then he is overbidding conditional on defeating only an opponent with a value-signal less than his own. A critical point occurs where nullclines intersect.
Assumption 1 ensures that the analysis of system (8) is routine. Figure 3 illustrates a representative case. Six facts are noteworthy.
(i) The curve ν(s) runs from the origin to the figure's top-right corner. In the region below ν(s),ḃ < 0; otherwise,ḃ > 0. Figure 3: A phase-portrait of system (8) with one critical point.
(iii) There exists at least one critical point when w = 0 (Lemma A.2 in Appendix A). For simplicity, Figure 3 shows an instance with one critical point at (s 0 , b 0 ). We address the case of multiple critical points in Online Appendix B. Multiple critical points do not affect Theorem 1, though an extended analysis is needed to identifyb(·).
(iv) The ratioḃ/ṡ is positive at point (s, b) only when b is "in between" ν(s) and ψ(s).
These are regions R 1 and R 2 in Figure 3. In these regions the solutions of (8) can be identified with increasing functions of s. Elsewhere, solutions have negative slope and, therefore, cannot constitute part of an admissible solution forb(s).
(v) Generically, each critical point is either a saddle point or a node (Lemma A.3). When there is one critical point, it is a saddle point.
(vi) When there is one critical point, of particular interest are the stable manifolds that approach it from below and above in regions R 1 and R 2 . (The Hartman-Grobman theorem implies that these manifolds are generically unique.) In Figure 3, these are the bold curves connecting (s 0 , b 0 ) with the boundaries. Henceforth, we use b * (s) to denote the union of these solution paths' closures as a function of s. The function b * (s) is strictly increasing, continuous, and an integral curve of (7).
We are now ready to identify the functionb(·). This function must be strictly increasing, continuous, and an integral curve of (7). When there is one critical point, as in Figure 3, only the stable solutions of (8) approaching the critical point-the function b * (s)-satisfy the stated requirements. And so, in this case, we defineb(s) ≡ b * (s) for all s.
The identification ofb(s) illustrates the competing incentives facing a high-budget bidder. Consider again Figure 3. Whenb(s) is confined to region R 1 , a type-s bidder would like to bid more to capitalize on his opponent's (possible) budget constraint. His incentive to do so is moderated by the fact that he is overbidding conditional on defeating an adversary with a value-signal less than his own, i.e.b(s) > v(s, s). In region R 2 , these incentives flip and v(s, s) >b(s).
A standard argument lets us confirm that Theorem 1 characterizes an equilibrium. 12 Proof of Theorem 1. Suppose bidder j adopts the strategy, β(s, w) = min{b(s), w} wherē b(s) satisfies the conditions stated in Theorem 1. (The preceding discussion has established the existence of this function.) The expected utility of bidder i of type (s, w) who bidsb(x) is given by (4). Differentiating (4), Hence, bidder i can increase his payoff by bidding less thanb(x). A parallel argument shows that when x < s, bidder i can increase his payoff by bidding more thanb(x). Together, the two cases imply that bidder i has no incentive to deviate from β(s, w) to another bid in the range ofb(·). 13 Instead, suppose bidder i bids outside the range ofb(·). Any bid strictly exceedingb (1) is dominated byb(1). The bid b ≤b(0) yields an expected payoff of To complete the proof, it is sufficient to show that ∂U(b|s, w)/∂b ≥ 0 when w ≤ b <b(0).
which implies the desired conclusion.
Remark 3. Theorem 1 generalizes to the case of affiliated value-signals . In this case, let h(s i , s j ) be the strictly positive, bounded, permutation-symmetric, and log-supermodular joint density of value-signals. 14 Define h(x|s) and H(x|s) as the conditional p.d.f. and c.d.f., respectively, and let λ(x|s) := h(x|s)/(1−H(x|s)). The analogue of Theorem 1 holds if v(·, x)λ(x|·) is nondecreasing, a common assumption (Krishna and Morgan, 1997;Lizzeri and Persico, 2000;Fang and Parreiras, 2002). This condition holds when, for example, v(s i , s j ) = (s i + s j )/2 and h(s i , s j ) = 4(1 + s i s j )/5. Intuitively, the restriction limits the informativeness of s i concerning s j relative to its impact on bidder i's own payoff. The supporting analysis is essentially identical to the preceding case and is omitted.
Remark 4. A closed-form expression forb(·) is unavailable, but the preceding analysis suggests how to approximate it numerically. First, identify a critical point in (8) through which the function must pass. The stable manifolds of system (8) approach this point with a slope equal to that of the negative eigenvector from the system's Jacobian matrix. An approximate linearized solution can be defined locally at the critical point. This solution can be extended to the domain's remainder with standard numerical methods.
Remark 5. Allowing for N > 2 bidders requires amending (4), the expression for an agent's expected payoff. Defeating an opponent with a low value-signal or a low budget implies different conclusions concerning the item's value. Each possibility needs to be accounted for on an opponent-by-opponent basis. Kotowski and Li (2014a) present this extension for the all-pay auction with interdependent values. The equilibrium with N bidders in that setting is qualitatively-similar to the two-bidder case. Che and Gale (1998) allow for N bidders and study canonical equilibria in the private-values case.

Non-Canonical Equilibria
A canonical equilibrium may not exist in the absence of Assumption 1 or if w > 0. To investigate this possibility, we focus on the case where w > 0 as the logical parallel with Example 1 is easiest to appreciate. A canonical equilibrium may not exist when w > 0 because the marginal return to a slightly higher bid changes appreciably at w. A bid above w defeats a budget-constrained opponent while a bid below w does not. This change in a bid's marginal return is particularly pronounced in Example 1 due to a mass point in the budget distribution at 1/4. As shown below, a mass point is not necessary to induce a similar jump discontinuity in the equilibrium strategy of a high-budget or unconstrained agent.
Consider the non-canonical strategy illustrated in Figure 4, which translates the intuition from Example 1 to a model with a continuous type-space. It has three parts. First, a bidder with a low value-signal bids less than w. Second, a high-budget bidder increases his bid discontinuously when his value-signal iss. Third, a low-budget bidder cannot match the jump in a high-budget bidder's strategy ats. Competition is stratified and low-budget bidders with value-signals s ∈ (s,s ′ ) compete only in a low range of bids. A novelty is that the distinction between high-and low-budget bidders is endogenous and changes with the agent's value-signal. In Figure 4(a), this boundary is given by the functionφ(·). If w <φ(s), the agent's budget is "low" (conditional on his value-signal) and he bids less than w; otherwise, his budget is "high" and he bids above w. While an equilibrium with the above characteristics can arise when w > 0, its presence is not assured. For some parameters, a canonical equilibrium may apply instead. To characterize all possibilities, we extend Theorem 1 to allow w ≥ 0. Concurrently, we also strengthen Assumption 1 (see Remark 1 above) to simplify the resulting technical analysis.
Theorem 2 allows for both canonical and non-canonical equilibria. Which case applies depends on the economy's parameters. Two qualitatively distinct canonical equilibria are defined solely by part (b). The functionb(·) in a Type 1 canonical equilibrium is characterized only by (10c). 15 This class subsumes the preceding subsection's analysis. As in the preceding subsection, the functionb(·) coincides with the stable solutions of system (8), the function b * (·). Given Assumption 1 ′ , a necessary and sufficient condition for a Type 1 canonical equilibrium to exist is that b * (·) exists and has domain [0, 1]. Intuitively, such equilibria tend to arise when w is close to zero.
A Type 2 canonical equilibrium is characterized by (10b) and (10c). 16 For low values of s,b(s) = α(s), the equilibrium bid in the absence of budget constraints and the solution to (10b). 17 If there exists a value s α ∈ (0, 1) such that α(s α ) = w, thenb(·) makes a continuous transition into the range of bids above w ats = s α . For s >s,b ′ (s) is given by (10c). Given Assumption 1 ′ , a necessary and sufficient condition for a Type 2 canonical equilibrium is the This condition must hold for (10c) to have an increasing solution at (s α , w). Online Appendix B discusses Type 2 equilibria in further detail. Intuitively, such equilibria tend to arise when w is relatively large.
The third case corresponds to the non-canonical equilibrium sketched in Figure 4. In this case both parts (a) and (b) in Theorem 2 contribute to the equilibrium strategy's definition.
In this subsection's remainder we outline this equilibrium's construction. The associated proofs are relegated to Appendix B. A numerical example is presented in Section 3.3.
Construction of a Non-Canonical Equilibrium Paralleling Section 3.1, the following discussion presumes the existence of a single critical point. The cases of no or multiple critical points are presented in Online Appendix B.
We first introduce an assumption (henceforth maintained) precluding Type 1 or Type 2 equilibria. It assures the applicability of the non-canonical case given Assumption 1 ′ .
Recall that b * (·) defines the stable solutions to system (8) and α(·) is the first-price auction's equilibrium strategy in the absence of budget constraints.  is not defined for all s. 18 Assumption 2(b) implies (10c) does not have an increasing solution at (s α , w). In the private values case, i.e. v(s i , s j ) = s i , it holds when (1 − . Thus, a sufficient condition for Assumption 2(b) to be satisfied is that the budget distribution is relatively concentrated near w. In this light, Example 1 can be interpreted as a limiting case where w = 1/4 is a mass point of the budget distribution.
Next, we explain how to identifys,b(·),b(·), andφ(·). 19 To identifys, we first introduce The function µ(·) is labeled in Figures 6 and 7. Through each point along this curve passes a solution of system (8) that can be extended as an increasing, continuous function to the boundary. 20 The values is a solution toŪ whereŪ In words, a bidder with value-signals is indifferent between the bids α(s) and µ(s). Lemma B.1 in Appendix B shows that (11) has a solutions ∈ [s * , s α ] given the maintained assumptions. Givens,b(·) is defined as follows. For s <s,b(s) solves (10b). Of course, this implies b(s) = α(s) for s <s. For s ≥s,b(s) coincides with the increasing solution to system (8) passing through point (s, µ(s)), which assures (10c) is satisfied. Figure 6 illustrates this definition in two representative cases. In Figure 6(a), µ(s) = b * (s) andb(s) = b * (s) for all s >s. In Figure 6(b), µ(s) = ψ(s) andb(s) < b * (s) for all s >s. There is a jump discontinuity inb(·) ats.
The functionφ(·) defines the boundary between high-and low-budget bidders along which β(s, w) is discontinuous. Accordingly, it is defined by an indifference condition. A type-(s,φ(s)) bidder is indifferent between the bidsb(s) andφ(s). Thus, A subtle complication is that the definitions ofb(·) andφ(·) reference one another. Thus, a fixed point argument is needed to show that these functions exist with the stated properties.
To reinforce intuition forφ(·), recall Example 1. In that case, a high-budget bidder's strategy is discontinuous in a neighborhood ofs. A low-budget agent bid less than 1/4 when s ∈ (s,s ′ ), but his bid increases to 1/4 once s >s ′ . The functionφ(·) generalizes this sequence of jump discontinuities to a continuum type space.
Verification that Theorem 2 describes an equilibrium mimics the proof of Theorem 1 and is relegated to Appendix B.
Remark 7. Absence Assumption 1 ′ , the auction's equilibrium will depend on the model's specifics. When the function b → η(s, s) − δ(b, s) crosses zero multiple times, some crossings occur at values where the budget distribution is (relatively) concentrated. Around these values, a bidder has an incentive to increase his bid substantially as the marginal returns to a slightly higher bid rise significantly. Jump discontinuities around such values allowb(s), the bid of an unconstrained bidder, to "skip around" regions where (10c) lacks increasing solutions. Multiple discontinuities are possible if G(·) is sufficiently irregular. Despite similar Figure 7: The functionsb,φ, andb. The functionb has a jump discontinuity ats.
intuition, identifying the analogues ofφ(·) andb(·) becomes challenging. Our proof does not extend immediately as it relies on the concavity of G(·) for this step.

An Example
An example can illustrate both canonical and non-canonical equilibria.
Example 2. Suppose v(s i , s j ) = s i + s j and value-signals are uniformly distributed on the unit interval. In the absence of budget constraints, α(s) = s is the symmetric equilibrium strategy. Suppose budgets are distributed uniformly on [w, 2] and w = 0. Figure 8(a) shows b(s), ψ(s), and ν(s). There is one critical point at s 0 ≈ 0.105. Representative solutions from the associated plane-autonomous system and α(s) are graphed for context. Since values are interdependent, even a bidder with value-signal s = 0 bids a positive amount, i.e.b(0) > 0.
Assumption 2 holds when 0.15 < w < 0.27; thus, a non-canonical equilibrium exists. that value-signal increases his bid from 0.181 to 0.286-an increase of 58 percent! Thus, an agent's strategic response to an opponent's possible budget constraint can be quantitatively large. When w > 0.27, the equilibrium is again canonical, but of the Type 2 variety. The jump discontinuity inb(s) disappears.
To highlight the empirical implications, Figures 8(c) and 8(d) present histograms approximating the equilibrium bid distributions. In the canonical case, the peak occurs around b(0) ≈ 0.12. All bids below this value equal the bidder's budget. In the non-canonical case, the equilibrium bid distribution has few bids around w = 0.2. The equilibrium bid distribution would be uniform in the absence of budget constraints.

Discussion
The Second-Price Auction It is instructive to compare equilibrium bidding in first-and second-price auctions. In the case of private values, i.e. v(s i , s j ) = s i , the second-price auction has an equilibrium where each type-(s, w) agent bids min{s, w} (Che and Gale, 1998). Fang and Parreiras (2002) examine the second-price auction with interdependent values in a two-bidder setting. They identify an equilibrium where each bidder's strategy is , which is the equilibrium strategy identified by  in the absence of budget constraints. Whenb II (s) ≥ w,b II (s) solves the differential equation subject to the boundary conditionb II (1) = v. Sinceb II (·) is increasing, there exists a valuẽ s II such that s < (>)s II =⇒b II (s) < (>)w. Generally,b II (·) may be discontinuous ats II .
While the resemblance of (7) and (15) is striking, three differences between the firstand second-price auctions are noteworthy. First, the boundary condition forb II (s) is predetermined. Second, the term "η(s, s) − b" appears in the denominator of (15) instead of "η(s, s) − δ(b, s)." Third, the discontinuity that can arise in β II (s, w) ats II is qualitatively different from that identified in Section 3.2 above. There is no separation among high-and low-budget bidders in the second-price auction. A bidder with value-signal s >s II bids above w and a gap in the equilibrium bid distribution may arise. In contrast, the equilibrium bid  23 distribution in the first-price auction has connected support, even in a non-canonical equilibrium. This difference is due to the distinct pricing mechanisms. In the second-price auction, an agent bidding infinitesimally above the gap has no reason to reduce his bid. His probability of winning and his expected payment are unchanged if he bids a bit less. In the first-price auction, an agent bidding infinitesimally above a (hypothetical) gap can significantly reduce his expected payment without appreciably reducing his probability of winning. As a result, a gap in the equilibrium bid distribution cannot arise. 22 Che and Gale (1998,2006) show that the first-price auction revenue-dominates the second-price auction in an independent private values setting with financial constraints.  show that the first-and second-price auctions are revenue equivalent in the absence of budget constraints when valuations are interdependent but agents' types are independent. 23 They also prove that the second-price auction is revenue-superior when agents' types are affiliated. Taken together, these results imply that no revenue ranking exists between first-and second-price auctions allowing for both budget constraints and affiliated values. (2007) shows that the first-price auction has a unique monotone pure strategy equilibrium when bidders are ex ante symmetric, types are affiliated, and values are interdependent. This conclusion does not extend to a setting with budget constraints. Recall that in a non-canonical equilibrium, an agent discontinuously increases his bid at some value-signals. He does so conjecturing the other bidder does likewise.

Equilibrium Non-Uniqueness McAdams
In equilibrium, of course, this conjecture is correct but the construction introduces an implicit coordination problem. Different values at which the common bidding strategy "jumps up" may be compatible with different equilibria, as illustrated by the following example.
Example 3. Consider a first-price auction with two bidders. 24 Given (s In the absence of budget constraints, α(s) = 2s/3 is the symmetric equilibrium bidding strategy . 22 This reasoning does not apply when there are atoms in the equilibrium bid distribution, as illustrated by Example 1. In the example, the bid 1/4 is placed with positive probability in equilibrium. An agent bidding 1/4 would like to bid more but cannot due to his budget constraint. A bid slightly less than 1/4, in contrast, leads to a discrete fall in the probability of winning and is therefore not beneficial. A gap in the equilibrium bid distribution just below 1/4 results.
23 See also the discussion in Krishna (2002, Section 7.1). 24 Calculations pertaining to this example are presented in Online Appendix A.
Now suppose bidders face a common budget constraint: w i = w j = 1/3 with probability 1. 25 There are now two symmetric equilibria with strategies and β B (s) = 1/3.
In the "β A "-equilibrium, the agents adopt the no-budget-constraints equilibrium strategy when s is small. Ats A = 1/9 their bids increase discontinuously from 2/27 to 1/3. In the "β B "-equilibrium, the threshold value-signal above which agents bid their entire budget is s B = 0. The "β B "-equilibrium is revenue superior.

Comparative Statics
The countervailing incentives associated with budget constraints typically lead to ambiguous comparative statics. For example, consider a "tightening" of financial constraints where the budget distribution is skewed in the sense of likelihood ratio dominance toward a distribution favoring lower budgets. 26 In the second-price auction, this change inflates the bid of an "unconstrained" bidder (Fang and Parreiras, 2002). In the first-price auction, a bidder's response may go either way. There is now less incentive to bid aggressively because competition is reduced at higher bid levels. However, if budgets become more concentrated around particular values it is easier to exploit the budget dimension for strategic gain. On balance, a bidder may respond with a higher or lower equilibrium bid.
We document these phenomena in an example in Online Appendix C.
Reserve Prices Setting the optimal reserve price with budget-constrained bidders is challenging. Reserve prices screen bidders on both value and budget dimensions. Expected revenues may fall if bidders are disproportionately screened along the latter as the probability of sale is particularly impacted. If agents are unlikely to be budget constrained, imposing the no-budget-constraints revenue-maximizing reserve price can be a sensible policy. For example, if in the model of Example 1 an agent has a low budget (w = 1/4) with probability p < 0.317, the revenue-maximizing reserve price is 1/2. This is also the revenue-maximizing reserve price in the absence of budget constraints. Otherwise, the optimal reserve price is less than 1/4. 25 We interpret this example as a limiting case of the more general model developed above. Posit, for instance, that w = 1/3 is the lower limit of the budget distribution's support and its density becomes highly concentrated around this value.
26 Likelihood ratio dominance implies hazard-rate dominance and first-order stochastic dominance.

Asymmetric Auctions and "Monotone" Equilibria
All equilibria in Section 3 are in nondecreasing strategies. A bidder with a higher budget or value-signal bids more. A natural question concerns this relationship's robustness, especially if budgets interact with preferences. For instance, consider risk preferences. A plausible claim is that a high-budget bidder is less risk averse than a low-budget rival, all else equal. A highbudget bidder may want to leverage his strategic advantage by bidding more. However, his higher risk tolerance can simultaneously tempt him to bid less. 27 Together these conflicting interests can lead to an equilibrium that fails to be "monotone" in the term's traditional sense, as shown by the following example.
Example 4. Recall Example 1, but now suppose that a winning bidder of type (s i , w i ) receives a payoff of (s i − b i ) w i +1/4 . Payoffs following a loss are zero. 28 Admittedly stylized, these payoffs capture the idea that a high-budget bidder is less risk-averse. A high budget (w i = 3/4) implies risk-neutrality. A low budget (w i = 1/4) implies strict risk-aversion. Figure 9 sketches this auction's symmetric equilibrium strategy, This strategy is neither canonical nor monotone. A high-budget bidder's risk tolerance explains his low bid when s i is small. He exercises his strategic advantage to outbid a low-budget rival if s i is large.
The absence of a monotone equilibrium clouds our understanding of the first-price auction with private budgets in general circumstances. Nevertheless, we can obtain some insight by noting that a systematic relationship still exists between types and bids. In Example 4, if s i and w i rise sufficiently and together, agent i will bid more. Reny (2011) showed that such regularities can be used to redefine a nondecreasing strategy in games of incomplete 27 Intuitively, the insurance provided by a higher bid is less valuable (Krishna, 2002, p. 38). 28 Cox et al. (1988) consider a setting with a similar utility function, though without budget constraints. 29 We verify that the defined strategy is an equilibrium in Online Appendix A. information. The idea is to realign changes in agents' types with changes in their equilibrium actions. This argument applies to the first-price auction with private budgets as well.
Generalizing our model, suppose there are N bidders, each with a private type (s . Now, let s = (s 1 , . . . , s N ) be the vector of value-signals and s −i be the vector s with s i removed. Define w and w −i analogously. Each bidder i has a set of valid bids B i ⊂ R, which subsumes a reserve price r i ≥ 0 and includes a "null bid" ℓ i < min{w, r i } guaranteeing a loss in the auction. 30 The function Suppose preferences satisfy the following conditions whenever r i ≤ b i ≤ w i : (i) If bidder i wins the auction with the bid b i , his payoff isū i (s i , s −i , w i , b i ). The function u i is bounded, nondecreasing in (s i , s −i , w i ), strictly decreasing in b i , and differentiable.
(ii) If bidder i does not win the auction, his payoff is u i (w i ). The function u i is bounded, nondecreasing, and differentiable.

Points (i)-(iv) adapt assumptions from Maskin and Riley
is an increasing-differences condition, common to the literature. 31 Point (v) is new. It says that a bidder's money holdings (budget) and the item for purchase are not excessively strong complements-the minimal marginal utility of an extra dollar conditional on winning the auction is less than the maximal marginal utility of an extra dollar conditional on losing the auction.
Point (v) plays a role in the redefinition of a nondecreasing strategy. Reny (2011) pioneered this approach and, interestingly, we can draw on his analysis of a multi-unit auction with risk-averse bidders to study our problem. The key analogy is the following. An auction with private budgets can be interpreted as an unusual multi-item auction where the highest bidder wins the item for sale, as usual, and all bidders "win" a cash prize equal to their private budget. Noting this analogy, we follow Reny (2011) by introducing a new ordering of each agent's type-space, which we denote by ≥ i . Defining we say that (s Figure 10 sketches the greater-and less-than sets for type (s i , w i ). We call the function To interpret ξ i and the ≥ i ordering, it is helpful to temporarily ignore the "sup" and "inf" operations. In the quasilinear case, ξ i = 0 because ∂u i /∂w i = ∂ū i /∂w i . Hence, ≥ i reduces to the standard coordinate-wise order of R 2 and a ≥ i -nondecreasing strategy is nondecreasing in the usual sense. A large value for ξ i can arise in two cases. If the numerator of (16) is large, then the marginal utility of an extra dollar is greater conditional on a loss than a win. All else equal, a bidder is more willing to bid less as w i increases since the sting of a loss is less severe. A ≥ i -nondecreasing strategy does not mandate a higher bid in this case. Alternatively, if the denominator of (16) is small, then payoffs are relatively unresponsive to changes in a bidder's value-signal. Thus, a (relatively) large increase in s i is necessary to assuredly counteract the (possible) countervailing incentives associated with an increase in w i . To ensure that ξ i is uniformly defined, the "sup" and "inf" operations pick out extreme Figure 10: The ≥ i ordering.
instances of the preceding cases. These help bound changes in a bidder's utility and allow us to confirm that a key step (Lemma C.1) in the proof of Theorem 3, which is stated below.
Concerning the information structure, we allow for affiliated value-signals, though qualified in the sense of Remark 3. Defining the terms we assume that the following conditions hold: (vi) Value signals are affiliated with a strictly positive and bounded joint density h(s). For (vii) Budgets are mutually independent and independent of value-signals with a strictly positive and bounded joint density. (The budget distribution may vary among bidders.) The two requirements in point (vi) ensure that the direct influence of s i on payoffs dominates its informational content concerning s −i . The condition is satisfied by the cases cited in Remark 3 above, which introduced a similar restriction.  ∞), and ties among high bidders are resolved with a uniform randomization.
The proof of Theorem 3 involves verifying that the model satisfies the conditions in Reny (2011). We already mentioned the analogy between our model and his analysis of a multiunit auction with risk-averse bidders. The key lemmas proceed similarly and quickly lead to part (a), which is the theorem's economically-salient conclusion. The restriction to finite and disjoint bid sets is a modeling device common in the auction literature, especially when types are multidimensional (for example, see McAdams, 2006).
The challenge in extending Theorem 3(b) to the interdependent-values case stems from economic, and not solely technical, considerations. It is instructive to see where standard arguments can fail. In a standard model, Athey (2001) and Reny and Zamir (2004) show that when an opponent adopts a nondecreasing strategy, a bidder benefits from bidding infinitesimally more to favorably resolve a hypothetical tie. This incentive is not assured in our setting due to the type-space's multidimensionality. 33 Defeating low-value-signal and low-budget opponents entails different inferences concerning the item's value. A bidder may hesitate to bid more when doing so disproportionately defeats low value-signal opponents. 34 The preceding results contrast with those of Gentry et al. (2015), who have adapted Reny's (2011) analysis to confirm the existence of a monotone equilibrium in a private-values first-price auction where agents' degree of risk aversion and wealth are private information.
Importantly, wealth is not associated with a constraint on bids in their model. Gentry et al. (2015) show that bids decline in wealth, due to agents' greater risk tolerance. This decline can occur in our setup, but the strategic option associated with budget constraints is a persistent countervailing force. A high-budget agent may wish to bid more to leverage his strategic advantage.

Concluding Remarks
We have examined the first-price auction under the assumption that agents face private budget constraints. Our analysis confirms that private budgets influence bidding in several for all s −i and s ′ −i . 33 Reny and Zamir (2004) provide additional examples illustrating the complications introduced by multidimensional types in auctions. 34 More generally, uniform tie-breaking is often inadequate to ensure equilibrium existence in bidding games with multidimensional private information (Araujo et al., 2008). Equilibrium existence can be restored by introducing endogenous tie-breaking rules (Simon and Zame, 1990;Jackson et al., 2002). In an earlier version of this study, a tie breaking rule proposed by Araujo and de Castro (2009) proved adequate for our model. distinct ways. They cap participants' bids while simultaneously encouraging more aggressive bidding in equilibrium by some agents. Together, these responses can lead to nontrivial equilibrium outcomes. Agents may adopt discontinuous bidding strategies and equilibria may fail to be monotone in the traditional sense of the term. Two directions for further study are especially noteworthy. First, we have precluded many natural embellishments from our model-resale, collusion, credit and financing, among others. As noted in Section 2, several studies have probed these questions, but much work remains. And second, the empirical implications of private budgets in auctions remain poorly understood. The development of new models or identification strategies to tackle this question seems especially promising.
Proof. Part (a) is immediate since δ(w, s) > v ≥ η(s, s). The proofs of (b) and (c) are similar and we only prove (b). To derive a contradiction, suppose ψ(s ′ ) =b > w. By We know that γ(b 0 ) ̸ = 0 and Assumption 1 implies that ∂δ ∂b Therefore, 4 ∂ṡ ∂b ∂ḃ ∂s > 0 as needed. Since the eigenvalues of J are real, by the Hartman-Grobman theorem, the critical point(s) will be a saddle or a node if the eigenvalues are non-zero. This is equivalent to J having a non-zero determinant at (s 0 , b 0 ): A perturbation of any of the distributions or of the valuation function is sufficient to ensure the preceding expression is non-zero.

B Proof of Theorem 2
The proof of Theorem 2 has two parts. The first part concerns the construction of a noncanonical strategy and complements Section 3.2. Construction of a Type 1 canonical strategy proceeds as in Section 3.1. Type 2 canonical strategies are discussed in Online Appendix B.
The second part verifies that the proposed strategy is a symmetric equilibrium.

B.1 Construction of a Non-Canonical Equilibrium Strategy
Outline The construction of a non-canonical equilibrium strategy with a discontinuity proceeds in several steps. First, Lemma B.1 confirms the existence of the values. Giveñ s, the definition ofb(·) is provided in Section 3.2 and Online Appendix B in other cases of interest. Then, we examine the functionsb(·) andφ(·). These functions are defined with reference to a fixed point of a particular operator, which is introduced in Remark B.4. Definitions B.1-B.4 and Lemma B.2 introduce preliminary concepts used in this definition.
Proof. We first make two observations. Recall that by convention G(b) = 0 if b < w.
The above cases exhaust the possibilities; hence, for all s ∈ [s,s 1 ),∆(s,φ(s)) = 0. By continuity, the preceding conclusion applies to the boundary s =s 1 as well.
Proof. The equality follows immediately from Remark B.5. That the derivative is greater than zero when evaluated atφ(s) follows from the definition ofφ(s). The valueφ(s) is either the smallest solution of φ →∆(s, φ) or this function's maximizer. As the function is concave, its derivative must be nonnegative atφ(s) whenφ(s) > w.
Remark B.6 (Definition ofs ′ ). Above we definedb,F andφ on the interval [s,s 1 ]. Ifb(s 1 ) < w then the preceding argument can be repeated inductively withs 1 replacings,b(s 1 ) replacing α(s),φ(s 1 ) replacingμ = µ(s), andF (s 1 ) replacing H(s), etc. Let [s,s ′ ] be the maximal interval on which the functionsb,F andφ can be defined inductively in this manner. The construction stops only in the limit. As the functionsb,F andφ are monotone, they can be extended tos ′ by taking the appropriate limit from the left.

B.2 Verification of Equilibrium
Let β be the strategy described in Theorem 2. We will verify that an agent does not have an incentive to deviate to an alternative feasible bid in the range of β given that the other agent bids according to β. In a type 1 canonical equilibrium, the proof of Theorem 1 applies. In a type 2 canonical equilibrium, the range of β equals the range ofb(·) and the argument below applies. (a) For all s <s,b(s) =α(s), the equilibrium bidding strategy in a first-price auction without budget constraints. Therefore, this bidder has no profitable deviation to any alternative bidb(x) where x <s . By continuity, the bidb(s − ) is also not a profitable deviation. Differentiating with respect to x and substitutingb ′ (x) as defined in (10a) gives , w). Therefore, the bidb(x) is not a profitable deviation.
(d) If this agent bidsb(x) where x ∈ (s, 1], his payoff is (B.4) Since s < x, by the same argument as in the proof of Theorem 1, we know that U(b(x)|s, w) ≤ U(b(s + )|s, w). Sinceb(s + ) =φ(s), the proof of sub-case (c) implies that U(b(s + )|s, w) ≤ U(b(s)|s, w). Therefore, the bidb(x) where x >s is not a profitable deviation.
Sub-cases (a)-(d) have shown that a bidder with value-signal s <s cannot gain by deviating to another bid in the range of β.
Case 2. Consider a type-(s, w) bidder where s ∈ (s,s ′ ) and w <φ(s). This bidder's expected payoff when bidding β(s, w) =b(s) is Deviations to bids of (a)b(x) for x <s or (b)b(x) for x ∈ [s,s ′ ) can be shown not to be profitable using an argument paralleling those presented in Cases 1(a) and 1(b) and Theorem 1. Thus, we only consider in the two remaining sub-cases, (c) and (d)

40
The first inequality follows fromφ ′ (x) ≤ 0 and η(s, x) ≥ η(x, x). The second inequality follows from Lemma B.5, which shows that the term in square brackets is nonnegative. Thus, each of these bids is dominated byφ(s w ) = w.
If s ≤s ′ , then U(φ(s w )|s, w) ≥ U(φ(s)|s, w) = U(b(s)|s, w) and the same argument as in Case 2 shows that this bidder has no profitable deviation to any bid below φ(s) as well.
Similarly, if s ≥s ′ , then U(φ(s w )|s, w) ≥ U(w|s, w). Verifying that bids below b(s ′ ) = w are not profitable deviations proceeds similarly to the argument in Case 1, sub-cases (a) and (b). A deviation to the bidb(x) where x >s is not feasible.
The same argument as in the proof of Theorem 1 shows that this agent has no profitable deviation to any bid in the range ofb(x) for x ≥s. In particular, this implies that he prefers β(s, w) to the bidb(s − ) =φ(s). And so, the same argument as in Case 3 above shows that he cannot gain by deviating to a bid less thanφ(s).
The preceding cases have shown that a type (s, w) bidder does not have a feasible, profitable deviation from β(s, w) given that the other bidder also adopts this strategy.

C Proof of Theorem 3
The proof of Theorem 3 follows a standard argument, building on Athey (2001) and Reny and Zamir (2004). We rely on Reny (2011) to confirm the existence of an equilibrium in ≥ i -nondecreasing strategies in a finite-action setting. This argument is similar to his study of a multi-unit auction. To simplify notation, in this appendix we let g(w) := i g i (w i ) be the joint p.f.d. of agents' budgets. Abusing notation, let g(w −i ) := j̸ =i g j (w j ). We write θ i = (s i , w i ) for agent i's type and the joint p.d.f. of agents' types is f (θ) := h(s)g(w).
Define ϕ i (b i , b −i ) as the probability that bidder i wins the auction with bid b i given that In the absence of ties, when others' strategies are β −i , bidder i's expected payoff at bid b i given θ i = (s i , w i ) is for all admissible strategies β i . When convenient, we use capital letters to refer to signals/types as random variables.
The following are preliminary lemmas. Lemma C.1 shows that 2 verifies a single crossing condition.
Proof. Reny (2011) provides a similar argument in the context of a multi-unit auction with risk averse bidders. Since we can establish the following inequalities: Rearranging the final inequality gives the desired result.
Lemma C.2. Let β −i be a profile of ≥ i -nondecreasing strategies for bidders j ̸ = i. Let θ i ≥ i θ ′ i and suppose the bids b i > b ′ i tie with probability zero with the bids of the other agents.
Proof. We prove part (a). Part (b) follows similarly. It is sufficient to show that First, define the following subsets of Θ −i : Then, .

Consider term [1]. By assumption
is nondecreasing in s −i when w −i is fixed. Hence, Next, consider term [2]. First, The second inequality follows from u(s i ,

Similar reasoning applies to term [3]:
The second inequality follows sinceH(s −i |s ′ i )/H(s −i |s i ) ≤ 1. And, the third is true be- Next, consider a finite-action variant of the model incorporating three modifications. First, restrict each agent's set of admissible bids to a finite set where r i ≥ 0 and ℓ i < r i . Assume that ℓ i ∈ B i and ∀i ̸ = j, B i ∩ B j = ∅. Therefore, relevant ties among bidders are not possible. Second, modify each bidder's utility function so that he incurs a penalty if he bids more than this private budget irrespective of the auction's outcome. 36 Suppose that this penalty is sufficiently large to ensure that all bids exceeding w i are strictly dominated by the bid ℓ i for a type-(s i , w i ) bidder. Third, letε > 0 be small and restrict each agent with a value-signal of s i ≤ s i +ε to bid ℓ i . This technical restriction is used by Athey (2001) and Reny and Zamir (2004) and its role here is the same. (Similarly, below we letε → 0.) Among other considerations, it ensures that all bids b i > ℓ i can win the auction with positive probability. All bids aboveB are now strictly dominated by the bid ℓ i .
In this modified game, let ρ i (θ i |β −i ) be the set of bidder i's best-response bids when his type is θ i and others follow the strategy profile Lemma C.3. Consider the modified auction game described above. Fix a ≥ i -nondecreasing strategy profile β −i . The set ρ i (θ i |β −i ) is nondecreasing in the strong set order.
Remark C.1. As the interim best reply correspondence is nondecreasing, player i has at least one nondecreasing best reply when others are following nondecreasing strategies (Topkis, 1998, Theorem 2.8.3).
Lemma C.4. There exists an equilibrium in ≥ i -nondecreasing strategies in the modified auction game described above.
Proof. The conclusion follows from Reny (2011) and specifically his Proposition 4.4, Theorems 4.1 and 4.3, and Remark 1. His conditions G.1-G.6 are satisfied by our game and are easy to verify.
Proof. Our proof builds on similar arguments provided by Athey (2001) and Reny and Zamir (2004), among others. The argument is simpler as ties are explicitly precluded. Consider a sequence of auctions indexed by k where the measure of agents constrained to bid ℓ i becomes arbitrarily small, i.e.ε k → 0. Let β k be the corresponding profile of equilibrium strategies. As each β k i is ≥ i -monotone there exists a strategy β i such that β k i (θ i ) → β i (θ i ) for all i and almost every θ i . For each i, β i is ≥ i -monotone. We will verify that β is an equilibrium whenε = 0.
Suppose the contrary and that bidder i has a profitable deviation from β i toβ i when β −i is the strategy profile of the other bidders. Thus, this convergence is uniform except possibly on a set of arbitrarily small measure (Egorov's Theorem). Choose ε > 0 sufficiently small such that 2ε sup u i (s, w i , b) < ∆/3. Let Θ ε be the set of measure (at most) ε where convergence need not be uniform. Next, consider On the set where convergence is uniform, Θ −i \ Θ ε −i , for k sufficiently large β k = β since the range of β k is a finite set. Thus, for k sufficiently large, for k sufficiently large. But, this implies that for k sufficiently large, which is a contradiction. (The middle term is nonpositive since β k i is a best response bid.) To prove Theorem 3(b) we consider a sequence of discretized auctions with increasingly finer sets of admissible bids. Athey (2001) and Reny and Zamir (2004), among others, also consider such sequences. Specifically suppose that in auction k, bidder i's set of feasible bids {ε i } are small fixed constants ensuring that P k i ∩ P k j = ∅ for all i, j, k. Contemporaneously, takeε k → 0 as well. Let β k be a corresponding sequence of equilibria.
Lemma C.6 (Theorem 3(b)). Suppose bidders have private values, ∞), and ties among high bidders are resolved with a uniform randomization. There exists an equilibrium in ≥ i -nondecreasing strategies.
Proof. This is a consequence of better-reply security (Reny, 1999); see especially Reny (2011, Corollary 5.2), whom we follow, and also Jackson and Swinkels (2005). Briefly, for k very large β k is an ϵ-equilibrium of the first-price auction where all bidders can place bids from B i . Given any ≥ i -nondecreasing strategy profile β −i , bidder i can approximate the payoff he would receive from the (feasible) strategy β i : Θ → B i . Specifically, suppose bidder i bids As k → ∞, the ex ante loss to bidder i from usingβ k i versus β i becomes arbitrarily small and this conclusion holds versus all strategies that the other bidders may follow.

A Examples
This section presents calculations pertaining to Examples 1, 3, and 4 from the main text.
is a symmetric Bayesian-Nash equilibrium.
Proof. The proof mirrors analogous arguments in the absence of budget constraints (for example, see Krishna, 2002, pp. 17-18). To simplify notation, letβ(s) := β(s, w) and β(s) := β(s, w). Let U(b|s, w) be the expected utility of bidder i of type (s, w) when he bids b and the other bidder follows β(s, w). We proceed to rule out deviations by each type of bidder. It is sufficient to rule out deviations to other bids in the range of β(s, w).
(c) Given β, the bid w is placed with positive probability by bidder j. If s ≤ w, bidding w cannot benefit bidder i. Else, if s > w, the bid w + ϵ gives bidder i a strictly higher payoff than w. Thus, sub-case (d) immediately below applies.
Therefore, this deviation is not profitable.
(c) Since w <s < s, the bidder can strictly improve upon the bid w by bidding ϵ more. Thus, the conclusion from sub-case (d) immediately below applies. Therefore, this deviation is not profitable.
Case 3. Consider bidder i of type s ∈ [0,s] with budget w = w. By identical arguments to those presented above, this bidder will not have a profitable deviation to any bid, except possibly w. This deviation yields a payoff of U(w|s, w) =
Therefore, this deviation is not profitable.
(c) A deviation to w can be ruled out by an argument similar to that from case 3.
The remaining candidate deviation is to the bid β( It is sufficient to establish that ∆(x, s) ≥ 0 for all x ∈ [s ′′ ,s ′ ) and s ≤s. Thus, a high-budget bidder with a value-signal of s >s has no profitable deviation fromβ(s).
Case 3. Consider bidder i with budget w = 1/4 and value-signal s ∈ [0,s ′′ ]. The bid β(x), x ∈ [0,s ′′ ], defeats a low-budget opponent with a value-signal less than x and a high-budget opponent with a value-signal less than 4x/3. The expected payoff is Maximizing the above expression with respect to x (subject to x ≤s ′′ ) gives a solution of x = s. Thus, β(s) is the best bid among bids in the range [0, 2s ′′ /3].
Case 4. Consider bidder i with budget w = 1/4 and value-signal s ∈ (s ′′ ,s ′ ]. By an argument Rearranging terms gives the desired conclusion.
As we have exhausted all possibilities, the proposed strategy is a symmetric equilibrium.

B Identification ofb(s)
A key step in the construction of an equilibrium strategy is the identification ofb(s), the bid of an high-budget or unconstrained agent. The main text outlines its identification when the

B.1 Multiple Critical Points and Canonical Equilibria
Consider the case in Figure B.1, which features three critical points and w = 0. When there are three critical points, they alternate between saddle points and nodes. Define b * (s) as a function coincident with the stable manifolds approaching the two saddle points from below and above. In Figure B.1, this is the bold curve. This function begins in region R 1 passes through (s 0 , b 0 ) and continues on to (s 1 , b 1 ). On the other side of (s 1 , b 1 ), b * (s) continues in region R 3 through (s 2 , b 2 ) and onto region R 4 . Note that b * (s) is defined for all s. As in the baseline case, we again identifyb(s) with b * (s). Further analysis of the canonical equilibrium is unaffected. The construction generalizes via induction to more critical points.

B.2 Multiple Critical Points and Non-Canonical Equilibria
A similar argument to the preceding case applies to non-canonical equilibria. A particular case with three critical points is illustrated in Figure B

B.3 No Critical Points and Non-Canonical Equilibria
When w > 0, system (B.1) may lack a critical point. In the absence of a critical point there is no "b * (·) function" and Assumption 2 reduces to only its second part: If α(s α ) = w for some s α ∈ (0, 1), then η(s α , s α ) − δ(w, s α ) > 0. An amendment of the preceding analysis allows us to identifyb(·). First, let

B.4 Type 2 Canonical Equilibria
When w > 0, there may exist a canonical equilibrium β(s, w) = min{b(s), w} whereb(s) is a continuous function. Analysis of this case is presented in Figure B.4, which assumes the existence of one critical point. The same analysis applies if there are no critical points or multiple critical points. Whenb(s) is less than w,b(s) = α(s), the equilibrium strategy in the first-price auction in the absence of budget constraints. If there exists s α ∈ (0, 1) such that α(s α ) = w, thenb(s) makes a continuous transition into the range of bids above w in a type 2 canonical equilibrium. The functionb(s) follows the increasing solution of system (B.1) starting at point (s α , w). For an increasing solution to exist at (s α , w), a necessary condition is that η(s α , s α ) − δ(w, s α ) ≤ 0. Note thats = s α andb(s) = w = α(s α ). Figure  B.5 illustrates a type 2 canonical equilibrium.

C Comparative Statics
The following example shows that skewing the budget distribution may induce higher or lower equilibrium bids by some unconstrained bidders.
Example C.1. Suppose there are two bidders with uniformly distributed value-signals on the unit interval. Suppose v(s i , s j ) = s i +s j . Consider three different budget distributions on [0, 2]: G 0 (w) = w/2, G 1 (w) = w/2, and G 2 (w) = w(4−w)/4. Each satisfies Assumption 1 ′ . The distribution G 0 likelihood ratio dominates G 1 and G 2 . Given each budget distribution there is one critical point. Given the budget distribution G 0 , the critical point occurs at  to G 2 induces a higher bid by an unconstrained bidder with a value-signal s ≈ 0.1.