Multi-bidder First Price Auction with Beliefs

Abstract

This paper characterizes the set of equilibria in the first price auction with multiple bidders—specifically three bidders, each of whose type space is multi-dimensional, incorporating a bidder’s beliefs about others’ valuations. In this auction, each bidder independently and privately learns the other two opponents’ valuations with some probability. This paper derives closed form solutions for equilibrium bidding behaviours parameterized by the degree of information when the bidder has homogeneous beliefs regarding each opponent. This paper demonstrates how much the level of information affects the bidding behaviours of the informed bidders. In addition, this paper extends the model into $N$ bidders when a high value bidder has fixed beliefs that all other bidders are identical types, and show how the value of information changes as the number of participants increases. Finally, this paper speculates on the possible changes in the efficiency of the model from increasing the valuation space to an arbitrary number.

JEL Classification: D44, D82, D83

Keywords

First price auction information level bidder’s beliefs multidimensional type space

Introduction

The standard assumption of private value auctions that bidders only observe their own valuations but have no further information regarding an opponent’s valuation, other than the distribution from which the valuation is drawn, is unrealistic. Bidders may have private observations, perfectly or imperfectly with noise, of their opponents’ valuations. As noted by Bergemann, Brooks, and Morris (2017), these observations are largely influential to the bidders’ optimal bidding strategies since information about others’ private values will set the boundaries of bidding supports for equilibrium. This is more significant in a first price auction (FPA) environment because an equilibrium bidding strategy in a FPA depends on a bidder’s beliefs about the valuations of other bidders. Based on this motivation, this paper studies a model of multiple-bidder FPA and explores the equilibrium bidding behaviours among bidders who are affected by information. Through this research, this paper analyses the marginal value of information as more information becomes available to the bidders.

In this paper, I consider a multi-dimensional private value model with three bidders in which each bidder has a binary valuation and gains probabilistic, yet perfect information regarding the other bidders’ valuations. This paper contributes to the literature on auctions with information by introducing a novel construction regarding the information space and its structure: multi-dimensional type space and several informed bidders characterized by observation likelihood. I specifically define the informed bidders given any information level regarding each opponent and derive closed form solutions parameterized by the information level. By deriving the bidding intervals generated by each information level, it is possible to obtain the value of information a bidder may have. I explore how the bidding behaviours of an informed bidder change as more information is available. Moreover, throughout this paper, by computing the bidding thresholds for the bidders I demonstrate how much a bidder’s information contributes to the auction environment as the number of participants increases.

This model is closely related to Fang and Morris (2006), in that (a) each bidder has one of two discrete valuation types, (b) private information is added to a bidder’s type space and (c) an equilibrium analysis on multidimensional FPA is performed. However, regarding equilibrium analysis, Fang and Morris (2006) only present a two-bidder case, while this paper considers a three-bidder auction environment, followed by an $N$ -bidder case to analyse the equilibrium bidding behaviours based on the premise that bidders can have perfect observation of their opponents’ valuations with probability. On the other hand, in the model of Fang and Morris, bidders have imperfect information from a noisy signal. In terms of perfect information, this paper is related to Kim and Che (2004) who also notice the impact of a bidder’s information on an equilibrium bidding strategy in FPA. Based on information partition where bidders have perfect and public information within a subgroup, Kim and Che (2004) assume continuous valuation and compare the allocative efficiency as well as the expected revenue between FPA and SPA when the bidders have within-group perfect information. However, their model is based on a dichotomous structure of the information and there is no direct way to understand the bidders’ equilibrium bidding behaviours in mixed strategies. They only provide an equilibrium analysis with pure strategies for a FPA. This is because in their model the group sizes may vary, yielding an asymmetric auction structure. Unlike Kim and Che (2004), the present paper focuses mainly on the mixed strategies occupied by a partially informed bidder who may face a potential high type bidder.

As in Fang and Morris (2006), the typical type space of a Bayesian game in FPA models is no more than two-dimensional. In the current paper with three bidders, the type space for a bidder extends to three dimensions: own valuation, information on the first opponent and information on the second opponent. When there are two possible types, low or high,¹

While a continuous valuation may seem more realistic, a discrete valuation environment is actually more so; real-life auctions have a finite valuation choice since any monetary value for an object can only have finitely many values between 0 and some maximum. The binary valuation model can be utilized to simulate a real-life auction of finite valuations because a high value bidder could adjust his bid to match that of the second-highest value bidder in the FPA environment.

for a bidder’s valuation, the information type regarding each opponent is threefold (uniformed, low and high) because the opponent’s valuation is either low or high, and because this information may or may not be revealed. When there is no correlation between each information set regarding the first opponent and the second, there is a maximum number of

∥ 2 ∥ \times ∥ 3 ∥ \times ∥ 3 ∥

possible types for each bidder: two types for one’s own valuation, three for his belief regarding the first opponent and another three regarding the second opponent. Because only the distinct combination of the opponents’ types matters, but not the order for a bidding strategy,²

For example, it is identical for an informed high value bidder to face against (high type, low type) or (low type, high type) in that one of them is a high type bidder and the other a low type regardless of order.

there are only 12 types to consider out of the 18 total possibilities.

In equilibrium, low value bidders bid their true value regardless of their information about the other bidders. However, high value bidders, except the fully informed one who faces two low value bidders, randomize over bid intervals according to certain bid distributions. In this case, the equilibrium bidding strategy is varied depending on the level of information. A contribution of this paper is characterizing the complex relationship between bid distributions and information. Specifically, as there is more information available, the supports of the distributions from which uninformed bidders randomize decrease, whereas the supports of the distributions from which informed bidders randomize increase. When the bidders obtain more information regarding the other bidders’ valuations, the more likely they rely on the second distribution. This paper characterizes the relationship between bidding behaviours and information for a general model of $N$ bidders assuming a high value bidder has homogeneous beliefs of all other bidders’ valuations.

Beyond the two papers directly related to the main topic of this paper, there are other research works on information structure: Azacis and Vida (2015) suggest certain signal structures of information to improve bidders’ payoff in FPA. The study by Bergemann et al. (2017) specifically focuses on correlated private information in FPA. When information is correlated, the minimum winning-bid distributions can be constructed from each model’s information structure in which the lower bound is attained. Similarly, Bergemann and Morris (2013, 2016) show that the range of these can be described in general game settings by a correlated equilibrium under incomplete information. The work of robust mechanism design by Brooks (2013) also has this direction.

The Model

Three risk neutral bidders, $i = 1, 2, 3,$ participate in a first price sealed bid auction and bid for an object. Bidders have independent private values for the object, drawn from identical distributions. Bidder $i$ ’s valuation $v_{i}$ takes one of two values, low or high, $v_{i} \in V = {V_{l}, V_{h}},$ with prior distribution ${1 - p,$ $p} .$ Moreover, bidder $i$ perfectly learns³

Therefore, there is no noise regarding the opponent’s type in the model.

the other two bidders’ valuations

v_{- i} \in V = {V_{l}, V_{h}}

with certain probabilities. Let

α_{j} \in [0, 1)

be the probability with which bidder

i

learns regarding the valuation of each of his opponents

j \neq i \in {1, 2, 3}

. Later, the model further assumes that the learning probability is symmetric: for bidders

i, j, k

, the probabilities with which bidder

i

learns bidder

j

’s and bidder

k

’s are equal

α_{j}^{i} = α_{k}^{i}

and vice versa. This learning process is independent and private. Also, the whole auction structure is common knowledge and exogenous.

All bidders submit their bids simultaneously and a highest bidder wins the object. The winner $i$ pays his bid $b_{i}$ , and therefore his payoff is $v_{i} - b_{i}$ , while the other bidders earn zero. In the case of a tie between a high value and a low value bidder, it is resolved in favour of the high value bidder. Similarly, in case of a tie between high value bidders, I adopt the tie breaking rule suggested by Kim and Che (2004).⁴

A tie will be broken in favour of high value bidders when there are multiple highest bids. The tie breaking rule will be random with equal probability if there are multiple highest bidders with the same valuation.

In the model, each bidder has a three-dimensional type space. One is for a bidder’s own valuation ${v_{i}},$ another for the bidder’s information regarding the valuation of the first opponent ${v^{1}}$ and lastly, one for his information regarding the valuation of the second opponent ${v^{2}}$ . Because it is possible that a bidder does not have any information regarding the opponents’ types, any bidder $i$ should belong to the following three-dimensional type space:

(v_{i}, v^{1}, v^{2}) \in {V_{l}, V_{h}}_{i} \times {U, V_{l}, V_{h}}^{1} \times {U, V_{l}, V_{h}}^{2},

where ${U, V_{l}, V_{h}}^{j}$ is the information space revealed to a bidder who is either uninformed $(U)$ or informed of whether the first or the second opponent is a low value bidder $(V_{l})$ or a high value bidder $(V_{h}) .$ Consequently, there are 18 different types of bidders. Among the 18 types, because 6 types are identical to other types in terms of bidding structure, I end up with considering only 12 distinct types. They are six types of low value bidders and another six types of high value bidders. Under this environment, I first characterize the equilibrium bidding strategies of the case, in which there is no information on the other bidders’ valuation. This is because the solution process in this part is a necessary step to the full model with information.

Characterization of Equilibrium

Equilibrium Bidding Strategy without Information

First, assume that there is no information regarding the other two bidders’ valuations: $α_{j} = α_{k} = 0.$ Then the type space has only private valuations, that is, $(v_{i}, v^{1}, v^{2}) \in {V_{l}, V_{h}}_{i} \times {U}^{1} \times {U}^{2}$ . The bidding strategy for a low value bidder is to bid $V_{l}$ . Uninformed low value bidders do not bid more than $V_{l}$ because they do not want to win the auction and pay more than their values. Also, they do not want to bid below $V_{l}$ because the other bidders may bid slightly more than that and could win the object with a positive payoff. However, the bidding strategy for a high value bidder with no information depends on who he may face against and thus he randomizes.

Proposition 1. The equilibrium bidding strategy for a high value bidder with no information is to randomize over an interval $[\underline{b}, (1 - (1 - p)^{2}) V_{h} + {(1 - p)}^{2} \underline{b}],$ following a distribution function $F (b)$ in which

F (b) = {\begin{matrix} 0 & i f & b = \underline{b} \\ \sqrt{\frac{K / p^{2}}{V_{h} - b}} - \frac{1 - p}{p} & i f & \underline{b} < b \leq (1 - {(1 - p)}^{2}) V_{h} + {(1 - p)}^{2} \underline{b} \\ 1 & i f & b > (1 - {(1 - p)}^{2}) V_{h} + {(1 - p)}^{2} \underline{b} \end{matrix}} .

Although the formal proof is presented in the Appendix, deriving the bidding interval is an important step towards the main theorem presented in the next section. To derive the bidding interval of the proposition, I first conjecture that a high value bidder without information randomizes over a certain interval, and then I determine the interval’s domain and the distribution over it.

Conjecture 1. A high value bidder randomizes over an interval $[\underline{b}, b^{\circ}],$ follwing a distribution function $F (b)$ with $\underline{b} = V_{l} .$

The function $F (b)$ satisfies the standard properties of a cumulative distribution function. The necessary condition for equilibrium is that the high value bidder is indifferent throughout all of the supports of the distribution for an expected profit from the strategy. Hence, there is a constant $K$ such that⁵

The three-bidder model without information is extended to the case of $M$ -valuations as follows: $K^{(M)} = (V_{h} - b) {[\sum_{m = 1}^{M} p_{m} F^{M} (b)]}^{2}$ , with $\sum_{m = 1}^{M} p_{m} = 1.$

K = (V_{h} - b) [{(1 - p)}^{2} + 2 p (1 - p) F (b) + p^{2} F {(b)}^{2}] .

(1)

The first term ${(1 - p)}^{2}$ is the probability that the high value bidder faces two low value bidders, while the last term $p^{2} F {(b)}^{2}$ is the probability of facing two high value bidders. The case of facing one high value bidder and one low value bidder is given by $2 p (1 - p) F (b)$ . Rearranging and solving for $F (b)$ to get

F (b) = \sqrt{\frac{K / p^{2}}{V_{h} - b}} - \frac{1 - p}{p} .

(2)

I further conjecture that there is no mass point at $\underline{b}$ , that is, $F (\underline{b}) = 0.$ Then $\sqrt{\frac{K / p^{2}}{V_{h} - \underline{b}}} - \frac{1 - p}{p} = 0$ or $K = (V_{h} - \underline{b}) {(1 - p)}^{2} .$ Substituting $K$ into $F (b)$ to get

F (b) = \sqrt{\frac{(V_{h} - \underline{b}) {(1 - p)}^{2} / p^{2}}{V_{h} - b}} - \frac{1 - p}{p} .

(3)

Figure 1.

Source: Author’s simulation result with the parameter values.

The upper bound of the support should satisfy $F (b^{\circ}) = 1$ :

F (b^{\circ}) = \sqrt{\frac{(V_{h} - \underline{b}) {(1 - p)}^{2} / p^{2}}{V_{h} - b^{\circ}}} - \frac{1 - p}{p} = 1.

(4)

Hence, the upper bound $b^{\circ} = [1 - (1 - p)^{2}] V_{h} + {(1 - p)}^{2} \underline{b} .$

Proof. See Appendix for the proof of Proposition 1 $□$

The non-trivial bidding interval is $\underline{b} < b \leq [1 - (1 - p)^{2}] V_{h} + {(1 - p)}^{2} \underline{b}$ . In a simpler case of only two bidders, the bidding interval is $\underline{b} < b \leq p V_{h} + (1 - p) \underline{b} .$ The upper bound in the model of three bidders is elongated by $[(1 - p) - (1 - p)^{2}] (V_{h} - \underline{b}) = (1 - p) p (V_{h} - \underline{b})$ to a higher point $b^{\circ} = [1 - (1 - p)^{2}] V_{h} + {(1 - p)}^{2} \underline{b}$ as shown in the following example.

Example: The bidding interval $[\underline{b}$ $,$ $b^{\circ}]$ for a three-bidder model of $p = 1 / 2$ with $V_{h} = 2$ and $V_{l} = 1$ is $[1, \frac{7}{4}] .$ The interval of a two-bidder case is $[1, \frac{3}{2}] .$ Figure 1 shows the two distribution functions with the bidding thresholds.

Equilibrium Bidding Strategy with Information

Let us modify the model so that there exists information with positive probability regarding an opponent’s type. Bidder $i$ learns the valuation of bidder $j$ , but only with probability $α_{j} \in (0, 1) .$ This implies that by the probability $1 - α_{j} \in (0, 1)$ , bidder $i$ is uninformed of the valuation of bidder $j$ , and thus he would randomize his bid against a potential high value bidder. With the three-bidder environment, the model has a three-dimensional type space, $(v_{i}, v^{1}, v^{2}) \in {V_{l}, V_{h}} \times {U, V_{l}, V_{h}}^{1} \times {U, V_{l}, V_{h}}^{2},$ with 18 possibilities among which 12 distinct types are considered. Of these 12, the bidding strategies for 6 low value bidders remain the same as in the no-information case. All six types of low value bidders bid $V_{l}$ and an informed high value bidder who knows that he faces two low value bidders also bids $V_{l} .$ However, of the remaining five high value bidders, those who are informed of the valuation of any of the two opponents construct their optimal bidding strategies according to the level of information they have. Since two of the five cases each include one unknown valuation, they are not tractable as closed form solutions.⁶

These are (a) the high value bidder is uninformed of one of his opponents but knows that the other bidder’s valuation is high; (b) the high value bidder is uninformed of one of his opponents, but knows that the other bidder’s valuation is low.

Therefore, this paper considers only the following three distinct cases:

(a)

the high value bidder is uninformed of both of his opponents’ valuations,

(b)

the higher bidder knows both of his opponents are high value bidders, and

(c)

the high value bidder knows that his opponents have different valuations. Regarding bidder

i

’s two opponents

j

and

k,

these cases are parametrized by some information level

α :

${1 - α_{j}, 1 - α_{k}}$

${α_{j},$ $α_{k}}$

${α_{j}, 1 - α_{k}} o r$ ${1 - α_{j}, α_{k}}$

For the necessity of obtaining a closed form solution, I assume that $α_{j} =$ $α_{k} = α$ , implying a high value bidder has about the same amount of information regarding the other two opponents.⁷

For heterogeneous beliefs, the Equations (5), (6), (7) or equivalently (8), (9), (10) will be modified to:

$\frac{K^{'}}{V_{h} - b} = [(1 - p) + p (1 - α_{j}) F_{j} (b)] [(1 - p) + p (1 - α_{k}) F_{k} (b)]$

$\frac{N^{'}}{(V_{h} - b)} = [(1 - α_{j}) + α_{j} G_{j} (b)] [(1 - α_{k}) + α_{k} G_{k} (b)]$

$\frac{O^{'}}{(V_{h} - b)} = [α_{k} (1 - α_{j}) + α_{k} α_{j} G_{j} (b) + (1 - α_{k}) (1 - α_{j}) G_{k} (b) + α_{j} (1 - α_{k}) G_{j} (b) G_{k} (b)] .$

Not only are these equations unsolvable for closed form solutions but also, given a constant $K^{'}$ , $N^{'}$ or $O^{'},$ the equations should admit many combinations of possible bidding distributions. Thus, in the first equation, $F_{j} (b)$ and $F_{k} (b)$ are inversely related to each other, and for a fixed $K^{'}$ , a higher distribution value of one of the two opponents implies a lower distribution value for the other opponent.

Proposition 2. For an $α$ -equilibrium bidding strategy, an uninformed high value bidder randomizes over an interval $[\underline{b}, V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}}],$ following $F (b)$ and an informed high value bidder randomizes over an interval $[V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}}, V_{h} - \frac{{(1 - α)}^{2}}{{(\frac{1 - α p}{1 - p})}^{2}} (V_{h} - \underline{b})],$ following $G (b) .$ The distribution functions are

F (b) = {\begin{array}{l} 0 & i f b = \underline{b} \\ (\frac{1}{1 - α}) (\frac{1 - p}{p}) [\sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1] & i f \underline{b} < b \leq V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{% 1 - α p}{1 - p})}^{2}} \\ 1 & i f b > V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{1- α p}{1-p})}^{2}} \end{array}}

and

G (b) = {\begin{matrix} 0 & i f & b = V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} \\ \frac{1 - α}{α} [\sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2} (V_{h} - b)}} - 1] & i f & V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}} < b \leq V_{h} - \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} \\ 1 & i f & b > V_{h} - \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} \end{matrix}} .

I utilize the following conjecture to derive the bidding intervals of the proposition, and then I present a formal proof.

Conjecture 2. An uninformed high value bidder who faces either a high value bidder or a low value bidder randomizes over an interval $[\underline{b}, b^{\circ}]$ following a distribution function $F (b) .$ An informed high value bidder who knows he faces a high value bidder randomizes over an interval $[b^{\circ}, b^{*}]$ following a distribution function $G (b)$ .

I first construct the necessary conditions for the three types of bidders to support an equilibrium based on the following notion: an informed high value bidder facing a high type bids according to $G (b)$ with probability $α,$ and for the rest of probability $1 - α,$ the high value bidder facing a high type bids according to another distribution $F (b) .$ The support for the function $F (b)$ is lower than that for $G (b)$ as shown in the following because the high value bidder who is uninformed of the high type opponent bids in the zone of $F (b),$ while the bidder who is aware of the high type opponent bids in the zone of $G (b)$ and thus his bid should be higher.

The high value bidder will randomize over each support so that a bid from the support will make the identical payoff when the bid is drawn from either distribution $F (b)$ or $G (b)$ . Therefore, given an information level $α,$ the necessary conditions for the equilibrium regarding $F (b)$ and $G (b)$ are⁸

Related to the note 6, the two cases that include one unknown valuation are $\frac{L}{(V_{h} - b)} = (1 - p) [(1 - α) + α G (b)] + p (1 - α) F (b) [α G (b) + (1 - α)]$ and $\frac{M}{(V_{h} - b)} =$ $(1 - p) [α + (1 - α) G (b)] + p (1 - α) F (b) [α + (1 - α) G (b)] .$

If the high value bidder is uninformed of both of his opponents, he is indifferent among all the elements of supports of the distribution $F$ . Hence, there must be a constant $K$ such that

\frac{K}{(V_{h} - b)} = [{(1 - p)}^{2} + 2 p (1 - p) (1 - α) F (b) + p^{2} {(1 - α)}^{2} F {(b)}^{2}] .

(5)

If the high value bidder knows that both of his opponents are high value bidders, then there must be a constant $N$ such that

\frac{N}{(V_{h} - b)} = [{(1 - α)}^{2} + α (1 - α) G (b) + α (1 - α) G (b) + α^{2} G {(b)}^{2}] .

(6)

If the high value bidder knows that his opponents have different valuations each other, then there must be a constant $O$ such that

\frac{O}{(V_{h} - b)} = [α (1 - α) + α^{2} G (b) + {(1 - α)}^{2} G (b) + α (1 - α) G {(b)}^{2}] .

(7)

Rearranging the Equations (5), (6) and (7) to get

\frac{K}{V_{h} - b} = {[(1 - p) + p (1 - α) F (b)]}^{2} .

(8)

\frac{N}{V_{h} - b} = {[(1 - α) + α G (b)]}^{2} .

(9)

\frac{O}{V_{h} - b} = α (1 - α) + [α^{2} + {(1 - α)}^{2}] G (b) + α (1 - α) G {(b)}^{2} .

(10)

From the Equations (8), (9) and (10), the distribution functions are derived as follows

F (b) = \frac{1}{(1 - α)} [\sqrt{\frac{K / p^{2}}{V_{h} - b}} - \frac{1 - p}{p}]

(11)

G_{1} (b) = \frac{1}{α} [\sqrt{\frac{N}{V_{h} - b}} - (1 - α)]

(12)

G_{2} (b) = \frac{- [α^{2} + {(1 - α)}^{2}] \pm \sqrt{{[α^{2} + {(1 - α)}^{2}]}^{2} - 4 α (1 - α) (α (1 - α) - \frac{O}{V_{h} - b})}}{2 α (1 - α)} .

(13)

Because the two cases of high value bidders described in Equations (9) and (10) are mutually exclusive, the conjecture is verified using a set of conditions from Equations (11) and (12) and another set from Equations (11) and (13). To determine $K,$ $N$ and $O$ in these distribution functions, I conjecture that there is no mass point at the lower bound of each support, that is, $F (\underline{b}) = 0$ and $G (b^{\circ}) = 0$ .⁹

The assumption that there is no mass point at $b^{\circ}$ helps to pin down the value of the $G_{1} (b)$ in Equation (9) and thus Equation (11). The assumption is made to ensure that there will be no overlapping zones of the supports in the two distributions, and there is a smooth transition from one function to the next. As can be verified in the proofs of Appendix, the condition holds when $α = 0$ so that $G_{1} (b)$ degenerates.

Also, the upper bounds of the two supports should satisfy

F (b^{\circ})

= 1

and

G (b^{*}) = 1,

respectively.

Figure 2.

Source: Author’s simulation result with the parameter values.

First, for $F (b)$ , Equation (11) satisfies $F (\underline{b}) = \frac{1}{1 - α} [\sqrt{\frac{K / p^{2}}{V_{h} - \underline{b}}} - \frac{1 - p}{p}] = 0$ or $K = (V_{h} - \underline{b}) {(1 - p)}^{2}$ . Substituting $K$ into $F (b)$ to get

F (b) = \frac{1}{1 - α} [\sqrt{\frac{(V_{h} - \underline{b}) {(1 - p)}^{2} / p^{2}}{V_{h} - b}} - \frac{1 - p}{p}] = \frac{1}{1 - α} \frac{1 - p}{p} [\sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1] ​ .

(14)

The upper bound of the support for $F (b)$ should satisfy $F (b^{\circ}) = (\frac{1}{1 - α}) (\frac{1 - p}{p}) [\sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b^{\circ}}} - 1] = 1.$ Hence, $b^{\circ} = V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}} .$ For $G (b)$ , Equation (12) satisfies $G_{1} (b^{\circ}) = \frac{1}{α} [\sqrt{\frac{N}{V_{h} - b^{\circ}}} - (1 - α)] = 0,$ or $N = {(1 - α)}^{2} (V_{h} - b^{\circ})$ from the conjecture on the lower bound $b^{\circ}$ associated to the distribution $G$ . Replacing with $b^{\circ} = V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}}$ to get $N = \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}}$ . Therefore,

G_{1} (b) = \frac{1}{α} [\sqrt{\frac{N}{V_{h} - b}} - (1 - α)] = \frac{1 - α}{α} [\sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2} (V_{h} - b)}} - 1] ​ .

(15)

Figure.3. Alpha and Distributions $F$ and $G$ in the Three-bidder Model

Source: Author’s simulation result.

Note: The figure shows bidding distributions over different values of $α$ , when $V_{h} = 2$ and $V_{l} = 1$ together with $p = 1 / 2.$ The darker graphs are those with lower information level $α .$ The lighter ones are with higher information.

From the conjecture about the upper bound, $G_{1} (b^{*}) = \frac{1 - α}{α} [\sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2} (V_{h} - b^{*})}} - 1] = 1.$ Thus, $b^{*} = V_{h} - \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} .$

Proof. See Appendix for the proof of Proposition 2 

Figure 2 shows the bidding thresholds, $b^{\circ}$ and $b^{*}$ over different values of $α$ , when $V_{h} = 2$ and $V_{l} = 1$ , together with $p = 1 / 2.$ As $α$ increases, the threshold $b^{\circ}$ for the distribution $F (b)$ approaches the low valuation $V_{l} = 1,$ whereas the threshold $b^{*}$ for the distribution $G (b)$ approaches the high valuation $V_{h} = 2.$ This implies that given the fixed value $V_{l},$ the bid support for the distribution $F$ decreases. Likewise, one may expect the bid support for the distribution $G$ to increase as $b^{*}$ approaches $V_{h} .$ However, the bidding support for the distribution $G$ depends not only on $b^{*}$ but also on $b^{\circ}$ and when the information level increases, the support for $G$ expands mainly because $b^{\circ}$ decreases, causing the support of $F$ to contract. In short, as bidders obtain more information regarding other bidders’ valuations, the more they rely on the distribution $G$ and the high value bidders will randomize over a wider range of bidding intervals as more information is available.

This finding is confirmed by Figure 3 which demonstrates the relationship between $α$ and the distribution functions $F (b)$ and $G_{1} (b)$ for the model of $V_{h} = 2,$ $V_{l} = 1,$ and $p = 1 / 2.$ When $α$ increases, the supports of the distribution $F (b)$ decrease but those of $G_{1} (b)$ increase. The figure shows that the support of distribution $G$ widens as the information level increases, which in fact expands in both directions, with more weight on the left side towards $V_{l}$ because the lower bound of the support for the distribution $G$ decreases faster than the upper bound increases.

Figure 4 shows the two distribution functions, $F (b)$ and $G_{1} (b)$ with the information parameter $α = 0.5$ , when $V_{h} = 2$ and $V_{l} = 1,$ with $p = 1 / 2.$ According to the graph, an uninformed high value bidder randomizes over $[1,$ $1.55]$ following $F (b),$ while an informed high value bidder randomizes over $[1.55,$ $1.89]$ following $G_{1} (b) .$ For the second set of Equations (11) and (13), $F (b)$ remains the same as above, but the function $G_{1} (b)$ is replaced by $G_{2} (b)$ .

Proposition 3. An alternative $α$ -equilibrium bidding strategy for an uninformed high value bidder is randomizing over an interval $[\underline{b}, V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}}],$ according to $F (b)$ and an informed high value bidder is randomizing over an interval $[V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}}, V_{h} - \frac{α (1 - α) (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}}],$ following $G_{2} (b),$ where F(b) =

{\begin{matrix} 0 & i f & b = \underline{b} \\ (\frac{1}{1 - α}) (\frac{1 - p}{p}) [\sqrt{% \frac{V_{h} - \underline{b}}{V_{h} - b}} - 1] & i f & \underline{b} < b \leq V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{% 1 - α p}{1 - p})}^{2}} \\ 1 & i f & b > V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{% 1 - α p}{1 - p})}^{2}} % \end{matrix} %}

and

G (b) = {\begin{matrix} 0 & i f & b = V_{h} - \frac{% (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} \\ \frac{1 - α}{α} [\sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2} (V_{h} - b)}} - 1] & i f & % V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}} % < b \leq V_{h} - \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{% 1 - α p}{1 - p})}^{2}} \\ 1 & i f & b > V_{h} - \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2} %} % \end{matrix} %} .

with $Q = α^{2} - {(1 - α)}^{2} + 4 {(\frac{α (1 - α)}{\frac{1 - α p}{1 - p}})}^{2} \frac{(V_{h} - \underline{b})}{(V_{h} - b)} .$

With the conjecture $G_{2} (b^{\circ}) = 0,$ Equations (13) pins down the unknown constant $O = \frac{α (1 - α) (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} .$ Thus, $G_{2} (b) = \frac{- [α^{2} + {(1 - α)}^{2}] \pm \sqrt{Q}}{2 α (1 - α)}$ , with $Q$ defined in the proposition. Because $G_{2} (b^{*}) = 1,$ it follows $2 α (1 - α) = - [α^{2} + {(1 - α)}^{2}] \pm \sqrt{Q^{*}}$ where $Q^{*} = {[α^{2} - {(1 - α)}^{2}]}^{2} + 4 {(\frac{α (1 - α)}{\frac{1 - α p}{1 - p}})}^{2} \frac{(V_{h} - \underline{b})}{(V_{h} - b^{*})}$ . For the negative root of $Q^{*}$ , no solution is found. For the positive root, $2 α (1 - α) = - [α^{2} + {(1 - α)}^{2}] + \sqrt{Q^{*}},$ the parametrized solution is $b^{*} = V_{h} - \frac{α (1 - α)}{{(\frac{1 - α p}{1 - p})}^{2}} (V_{h} - \underline{b})$ for all values of $α \in (0, 1) .$

Proof. See Appendix for the proof of Proposition 3 

Comparison and Extension

The following two tables summarize the main finding. The tables also include the simplest two-bidder case for comparison. Table 1 shows the bidding distribution functions for the case of the two-bidder model and the three-bidder model. Table 2 shows the bidding intervals to support these.

Figure 4.

Source: Author’s simulation result.

One may notice the pattern of the distribution functions and the bid intervals over the number of bidders. For any number of bidders $N,$ it is possible to generalize the model for the case where a high value bidder $i$ knows with $α_{j},$ for all $j \neq i \in {1, 2, ..., N},$ that all other bidders are identical types. If his beliefs are homogenous regarding all opponents, $α_{j} = α$ for all $j$ , then a closed form solution is obtained.

Proposition 4.

A

(N + 1)

-bidder

α

-equilibrium bidding strategy for an uninformed high value bidder

i s

t o

randomize over

a n

i n t e r v a l

[\underline{b}, V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{n}}] ​,

following

a

distribution function

F (b)

a n d

a n

informed

h i g h

v a l u e

b i d d e r

who faces all high types

i s

t o

randomize

o v e r

a n

i n t e r v a l

[V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{n}}, V_{h} - \frac{{(1 - α)}^{n} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}}] ​,

f o l l o w i n g

a

d i s t r i b u t i o n

f u n c t i o n

G (b) .

The distribution functions are

Table 1.

Bidding Distributions

Distribution	Two-bidder Model	Three-bidder Model
$F$	$(\frac{1}{1 - α}) (\frac{1 - p}{p}) [\frac{V_{h} - \underline{b}}{V_{h} - b} - 1]$	$(\frac{1}{1 - α}) (\frac{1 - p}{p}) [\sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1]$
$G_{1}$	$\frac{1 - α}{α} [\frac{V_{h} - \underline{b}}{(\frac{1 - α p}{1 - p}) (V_{h} - b)} - 1]$	$\frac{1 - α}{α} [\sqrt{\frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2} (V_{h} - b)}} - 1]$
$G_{2}$		$\frac{- [α^{2} + {(1 - α)}^{2}] \pm \sqrt{{[α^{2} - {(1 - α)}^{2}]}^{2} + 4 {(\frac{α (1 - α)}{\frac{1 - α p}{1 - p}})}^{2} \frac{(V_{h} - \underline{b})}{(V_{h} - b)}}}{2 α (1 - α)}$

Source: Author’s derivation/computation from the model.

Table 2.

Bidding Intervals

Bidding Interval	Two-bidder Model	Three-bidder Model
$b^{\circ}$	$V_{h} - \frac{1}{(\frac{1 - α p}{1 - p})} (V_{h} - \underline{b})$	$V_{h} - \frac{1}{{(\frac{1 - α p}{1 - p})}^{2}} (V_{h} - \underline{b})$
$b_{1}^{*}$	$V_{h} - \frac{1 - α}{(\frac{1 - α p}{1 - p})} (V_{h} - \underline{b})$	$V_{h} - \frac{{(1 - α)}^{2}}{{(\frac{1 - α p}{1 - p})}^{2}} (V_{h} - \underline{b})$
$b_{2}^{*}$		$V_{h} - \frac{α (1 - α)}{{(\frac{1 - α p}{1 - p})}^{2}} (V_{h} - \underline{b})$

Source: Author’s derivation/computation from the model.

F (b) = {\begin{array}{l} 0 & i f b = \underline{b} \\ (\frac{1}{1 - α}) (\frac{1 - p}{p}) [\sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1] & i f \underline{b} < b \leq V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{% 1 - α p}{1 - p})}^{n}} \\ 1 & i f b > V_{h} - \frac{(V_{h} - \underline{b} %)}{{(\frac{1- α p}{1-p})}^{n}} \end{array}}

and

G (b) = {\begin{array}{l} 0 & i f b = V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{n}} \\ \frac{1 - α}{α} [n \sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n} (V_{h} - b)}} - 1] & i f V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{n}} < b \leq V_{h} - \frac{{(1 - α)}^{n} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}} \\ 1 & f b > V_{h} - \frac{{(1 - α)}^{n} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}} \end{array}}

Conjecturing $F (\underline{b}) = 0$ and $G (b^{\circ}) = 0$ on $\frac{K_{n}}{(V_{h} - b)} = {[(1 - p) + p (1 - α) F (b)]}^{n}$ and $\frac{N_{n}}{(V_{h} - b)} = {[(1 - α) + α G (b)]}^{n}$ returns $\frac{1}{p (1 - α)} [^{n} \sqrt{\frac{K_{n}}{(V_{h} - b)}} - (1 - p)] = 0$ and $\frac{1}{α} [^{n} \sqrt{\frac{N_{n}}{V_{h} - b}} - (1 - α)] = 0.$ ¹⁰

Therefore, $F (b) = \frac{1}{p (1 - α)} [^{n} \sqrt{\frac{K_{n}}{(V_{h} - b)}} - (1 - p)]$ and $G (b) = \frac{1}{α} [^{n} \sqrt{\frac{N_{n}}{V_{h} - b}} - (1 - α)]$ .

Thus,

K_{n} = (V_{h} - \underline{b}) {(1 - p)}^{n}

and

N_{n} = {(1 - α)}^{n} (V_{h} - b^{\circ})

to get

F (b) = (\frac{1}{1 - α}) (\frac{1 - p}{p}) [^{n} \sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1]

and

G (b) = \frac{1 - α}{α} [^{n} \sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n} (V_{h} - b)}} - 1]

. From

F (b^{\circ})

= 1

and

G (b^{*}) = 1,

the bidding interval is obtained by

b^{\circ} = V_{h} - \frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}}

and

b^{*} = V_{h} - \frac{{(1 - α)}^{n} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}} .

Figure.5. $N$ and Distribution $F$ and G

Source: Author’s simulation result.

Note: The figures show bidding distributions over various number of bidders with $α = 0.5$ , when $V_{h} = 2$ and $V_{l} = 1,$ together with $p = 1 / 2.$

Proof. See Appendix for the proof of Proposition 4 

Figure 5 shows the relationship between the numbers of bidders $N$ and the distribution functions $F (b)$ and $G (b)$ over various values of $N .$ According to these graphs, as $N$ becomes bigger, the supports of the distribution $F$ increase while those of the distribution $G$ decrease. This is because as $N$ increases, the threshold $b^{\circ}$ for the distribution $F$ approaches the high valuation $V_{h} = 2$ as $N$ increases. The threshold $b^{*}$ for the distribution $G$ approaches this as well. This implies that given the fixed value $V_{h}$ the bid support for the distribution $G$ decreases. However, the bid support for the distribution $F$ increases as $b^{\circ}$ approaches to $V_{h} .$ Thus, when the number of bidders increases, the bidders are more likely to rely on the distribution $F .$ The figure explains that as more people join the auction, the impact of information is lessened.

Concluding Remarks

The degree of information a high value bidder has access to is crucial to the bidder’s optimal bidding behaviour. Under the assumption that a high type bidder has equal beliefs regarding the opponents’ types, this paper derives closed form solutions for the equilibrium bidding behaviours parametrized by the degree of information. Given a fixed number of bidders, the bidder’s winning bid is increased as more information about potential high type bidders is available. With more information, the support of the distribution function designed for an uninformed high value bidder decreases, while that of the distribution of an informed high value bidder increases. This implies that the bidders not only tend to bid more aggressively but also utilize wider bidding ranges for mixed strategies.

Introducing more bidders with beliefs into a FPA environment would produce a more realistic bidding case. However, as the number of bidders increases, the supports of the distribution functions for an uninformed high value bidder increase, whereas those of the distributions of an informed high value bidder decrease. This implies that the value of information may be limited by the number of participants: the greater the number of participants, the less the value of information about high type opponents. As such, under this environment, the informed high value bidder’s utility decreases along with it.

Information and Revenue: Would the seller benefit from releasing the information on bidders’ types? In section ‘Equilibrium Bidding Strategy with Information’ when there is more information, $b^{*}$ increases while $b^{o}$ decreases. A more informed high value bidder has a larger range of bidding in his mixed strategy, implying the bidder may bid either higher or lower than with less information. Therefore, the seller’s revenue has a possibility either to increase or decrease when the information level increases; this depends not only on the information but also on the prior distribution of the valuation, as well as the valuation range itself. For the case of $p = 1 / 2$ in the three-bidder example, when the information level increases from $α = 0.5$ to $α = 0.8,$ the bidding range of the informed high value bidder changes from $[1.55,$ $1.89]$ to $[1.31,$ $1.97]$ . Because there is no general result of impact on the seller’s expected revenue, it can be inferred that the direction of information matters in determining whether or not the seller would want to release the information to increase revenue.

Comparing with second price auction (SPA), note that to the bidders, FPA with information is more beneficial than SPA because bidders in FPA can utilize information to their advantage. On the other hand, the FPA is worse for the seller’s revenue, since bidders take advantage of available information. Naturally, the revenue equivalence does not hold. Generally, in this model setting with information, the FPA offers the seller no greater expected revenue than the SPA in any equilibrium.

Multiple Valuation and Efficiency: This auction is efficient because high value bidders win. This efficiency largely comes from the model’s default structure of binary valuation and non-overlapping bidding supports. Increasing the valuation types of a bidder affects the efficiency of the model. A multi-valuation type FPA with non-overlapping supports would still be efficient, since it can be reduced to a slightly modified binary-valuation model of the paper. Equilibria of the FPA may be inefficient if the supports of different types of bidders overlap in such a way that the order of valuation does not match with the order of points in bidding intervals from an information structure. In a model of ternary valuation $(V_{l}, V_{m}, V_{h})$ , the equilibrium bidding support of an uninformed high value bidder can be lower than the support of an informed middle value bidder if the middle value bidder believes he is likely to face a high value bidder, while the uninformed high value bidder bids as if he faces against a low one. As the number of valuation types increases, there is a greater chance of inefficient allocation. For additional discussion of ternary (or more) valuation related to efficiency, interested readers may look at Kim and Che (2004).

Although the equilibrium in FPA with multiple valuations is likely to be inefficient in a general setting, or perhaps may not exist at all, there may exist systematic approaches which can analytically characterize the multi-valuation auctions with overlapping distributions. This is still an open question to explore in proper detail.

Appendix

Proof of Proposition 1. Suppose the other two bidders bid according to the strategies specified in the model. Then a high value bidder’s expected payoff from the suggested strategies is $(V_{h} - b) [{(1 - p)}^{2} + 2 p (1 - p) F (b) + p^{2} F {(b)}^{2}]$ for $b \in [\underline{b}, b^{\circ}],$ where $b^{\circ} = (1 - (1 - p)^{2}) V_{h} + {(1 - p)}^{2} \underline{b} .$ Plugging $F (b)$ into this equation yields a constant, which is $K$ since the expected payoff returns $(V_{h} - b) {[(1 - p) + p F (b)]}^{2} = (V_{h} - b) {[(1 - p) + p (\sqrt{\frac{K / p^{2}}{V_{h} - b}} - \frac{1 - p}{p})]}^{2} = K .$ Therefore, the uninformed high value bidder is indifferent between any bids in the interval $[\underline{b},$ $b^{\circ}]$ . To complete the proof, I need to check whether the bidder does not have an incentive to deviate: the uninformed high value bidder has no incentive to raise his bid above $b^{\circ} .$ When the bidder does raise his bid above $b^{\circ},$ his expected payoff $π (b)$ is strictly decreasing in $b$ because $π (b) = (V_{h} - b) [(1 - p)^{2} + 2 p (1 - p) + p^{2}] = (V_{h} - b)$ . Also, the bidder does not have an incentive to bid below $\underline{b} = V_{l}$ because, in this case, his expected payoff is strictly less than $K$ . If $b \in [\underline{b}, b^{\circ}],$ his expected payoff $π (b)$ is $K$ by construction. Thus, in any case, there is no profitable deviation. 

Proof of Proposition 2. Suppose the other two bidders bid according to the strategies specified in the model. Consider the first case in which a high value bidder is uninformed by $1 - α$ of both of the other bidders. Then the high value bidder’s expected payoff from the suggested strategies is $(V_{h} - b) [{(1 - p)}^{2} + 2 p (1 - p) (1 - α) F (b) + p^{2} {(1 - α)}^{2} F {(b)}^{2}]$ for $b \in [\underline{b}, b^{\circ}],$ where $b^{\circ} = V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}} .$ Plugging $F (b)$ into this equation yields a constant, which is $K,$ because the expected payoff returns $(V_{h} - b) {[(1 - p) + p (1 - α) (\frac{1}{1 - α}) (\frac{1 - p}{p}) (\sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1)]}^{2} = (V_{h} - \underline{b}) {(1 - p)}^{2} = K .$ Thus, the high value bidder is indifferent between any bids in the interval $[\underline{b}, b^{\circ}]$ . To complete the proof of this part, I check if the bidder has an incentive to deviate. The uninformed high value bidder would not bid below $\underline{b}$ because his expected payoff is strictly less than $K .$ Also, the uninformed high value bidder would not bid more than $b^{\circ}$ because his expected payoff $π (b) = (V_{h} - b) {[(1 - p) + p (1 - α)]}^{2}$ is strictly decreasing in $b .$ When $b \in [\underline{b}, b^{\circ}],$ his expected payoff is by construction. Therefore, there is no profitable deviation. Second, if the high value bidder knows with probability $α$ that both of his opponents are high types, his expected payoff is $K$ $(V_{h} - b) [{(1 - α)}^{2} + 2 α (1 - α) G (b) + α^{2} G {(b)}^{2}]$ for $b \in [b^{\circ}, b^{*}],$ where $b^{*} = V_{h} - \frac{{(1 - α)}^{2}}{{(\frac{1 - α p}{1 - p})}^{2}} (V_{h} - \underline{b}) .$ The first term ${(1 - α)}^{2}$ is the probability that the high value bidder does not know that he faces two low value bidders and the last term is the probability that the high value bidder faces two high value bidders. The middle term specifies the situation of a mixture. Plugging $G (b)$ into this yields a constant, which is $N$ because the expected payoff returns $(V_{h} - b) {[(1 - α) + α \frac{1 - α}{α} (\sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2} (V_{h} - b)}} - 1)]}^{2} = \frac{{(1 - α)}^{2} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} = N .$ I again check whether the bidder does not have an incentive to deviate. The informed high value bidder would not bid below $b^{\circ}$ because this will lower his expected payoff below $N : π (b) = (V_{h} - b) {(1 - α)}^{2} < (V_{h} - b) {[(1 - α) + α G (b)]}^{2} = N$ . On the other hand, the bidder would not bid more than $b^{*}$ because in this case he would get a payoff of $(V_{h} - b) {(1 - p) (1 - p) + 2 p (1 - p) [(1 - α) + α G (b)] + p^{2} {[(1 - α) + α G (b)]}^{2}}$ or $π (b) = (V_{h} - b) {[1 - p α + p α G (b)]}^{2},$ which is strict $G (b)$ ly decreasing in $b .$ When $b \in [b^{\circ}, b^{*}],$ his expected payoff is $N$ by construction. Therefore, the bidder would not deviate. 

Proof of Proposition 3. Suppose that the other bidders bid according to the strategies specified in the model. Because $F (b)$ remains the same as in Proposition 2, the uninformed high value bidder would not deviate from bids in the interval $[\underline{b}, b^{\circ}],$ where $b^{\circ} = V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{2}} .$ Next is the case in which the high value bidder knows with $α$ that one of his opponents is a high type. Then the high value bidder’s expected payoff is $(V_{h} - b) [α (1 - α) + α^{2} G (b) + {(1 - α)}^{2} G (b) + α (1 - α) G {(b)}^{2}]$ for $b \in [b^{\circ}, b^{*}]$ , where $b^{*} = V_{h} - \frac{α (1 - α) (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} .$ Plugging into this yields a constant $O,$ because the expected payoff returns $(V_{h} - b) α (1 - α) ({[\frac{[α^{2} + {(1 - α)}^{2}]}{2 α (1 - α)} - \frac{[α^{2} + {(1 - α)}^{2}] \pm \sqrt{Q}}{2 α (1 - α)}]}^{2} + 1 - (\frac{{[α^{2} + {(1 - α)}^{2}]}^{2}}{{(2 α (1 - α))}^{2}})) = \frac{α (1 - α) (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{2}} = O .$ To complete this, I check if the bidder has any incentive to deviate. The informed high value bidder would not bid below $b^{\circ}$ because this will lower his expected payoff below $O : π (b) = (V_{h} - b) α (1 - α) < (V_{h} - b) [α (1 - α) + α^{2} G (b) + {(1 - α)}^{2} G (b) + α (1 - α) G {(b)}^{2}] = O$ . On the other hand, the bidder would not bid more than $b^{*}$ because in this case he would get a payoff $π (b) = (V_{h} - b) [2 α (1 - α) + α^{2} + {(1 - α)}^{2}],$ which is strictly decreasing in $b .$ When $b \in [b^{\circ}, b^{*}],$ his expected payoff is $O$ by construction. Therefore, the bidder would not deviate. 

Proof of Proposition 4. Suppose that all the other $N$ bidders bid according to the strategies specified in the model. Consider the first case in which a high value bidder is uninformed of any of the other bidders. Then, the high value bidder’s expected payoff from the postulated strategies is $(V_{h} - b) [(\begin{matrix} n \\ 0 \end{matrix}) {(1 - p)}^{n} + ... + (\begin{matrix} n \\ n \end{matrix}) {[p (1 - α) F (b)]}^{n}]$ with $b \in [\underline{b}, b^{\circ}],$ where $b^{\circ} = V_{h} - \frac{V_{h} - \underline{b}}{{(\frac{1 - α p}{1 - p})}^{n}} .$ Each term represents the probability that the uninformed high value bidder faces $1$ through $N$ low value bidders. Plugging $F (b)$ into this equation yields a constant, which $b \in [b^{\circ}, b^{*}]$ $[\underline{b}, b^{\circ}]$ is $K_{n}$ because the expected payoff returns $(V_{h} - b) {[(1 - p) + p (1 - α) (\frac{1}{1 - α}) (\frac{1 - p}{p}) (^{n} \sqrt{\frac{V_{h} - \underline{b}}{V_{h} - b}} - 1)]}^{n} = (V_{h} - \underline{b}) {(1 - p)}^{n} = K_{n} .$ Therefore, the high value bidder is indifferent between any bids in the interval . The uninformed high value bidder would not bid below $\underline{b}$ because his expected payoff would be strictly less than $K_{n} .$ Also, the bidder would not bid more than $b^{\circ}$ because his expected payoff $π (b) = (V_{h} - b) {[(1 - p) + p (1 - α)]}^{n}$ is strictly decreasing in $b .$ When $b \in [\underline{b}, b^{\circ}],$ his expected payoff is $K_{n}$ by construction. Second, if the high value bidder knows with $α$ that all of his opponents are high types, the high value bidder’s expected payoff from the postulated strategies is $(V_{h} - b) [(\begin{matrix} n \\ 0 \end{matrix}) {(1 - α)}^{n} + ... + (\begin{matrix} n \\ n \end{matrix}) {[α G (b)]}^{n}]$ with where $b^{*} = V_{h} - \frac{{(1 - α)}^{n} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}} .$ Each term represents the probability that the high value bidder knows with $α$ that he faces $1$ through $N$ high value bidders. Plugging $G (b)$ into this equation yields a constant which is $N_{n},$ since the expected payoff returns $(V_{h} - b) {[(1 - α) + α \frac{1 - α}{α} (^{n} \sqrt{\frac{(V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n} (V_{h} - b)}} - 1)]}^{n} = \frac{{(1 - α)}^{n} (V_{h} - \underline{b})}{{(\frac{1 - α p}{1 - p})}^{n}} = N_{n} .$ The informed high value bidder would not bid below $b^{\circ}$ because this will lower his expected payoff below $N_{n} : π (b) = (V_{h} - b) {(1 - α)}^{n} < (V_{h} - b) {[(1 - α) + α G (b)]}^{n} = N_{n}$ . On the other hand, the bidder would not bid more than $b^{*}$ because in this case he would get a payoff of $π (b) = (V_{h} - b) {[1 - p α + p α G (b)]}^{n},$ which is strictly decreasing in $b .$ If $b \in [b^{\circ}, b^{*}],$ his expected payoff is $N_{n}$ by construction. Therefore, the bidder would not deviate in any case. 

Footnotes

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author received no financial support for the research, authorship and/or publication of this article.

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Multi-bidder First Price Auction with Beliefs

Abstract

Abstract

Keywords

Introduction

The Model

Characterization of Equilibrium

Equilibrium Bidding Strategy without Information

Equilibrium Bidding Strategy with Information

Figure.3. Alpha and Distributions F and G in the Three-bidder Model

Comparison and Extension

Bidding Distributions

Bidding Intervals

Figure.5. N and Distribution F and G

Concluding Remarks

Appendix

Footnotes

Declaration of Conflicting Interests

Funding

References

Figure.3. Alpha and Distributions $F$ and $G$ in the Three-bidder Model

Figure.5. $N$ and Distribution $F$ and G