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Value-Rationalizability in Auction Bidding

Abstract

In a symmetric first-price or second-price auction, a bidding strategy is characterized as value-rationalizable if it can be viewed as being realized at a Nash equilibrium under the specification of a non-negative and increasing value function. In an environment where the underlying probabilistic framework is common knowledge, we investigate conditions for value-rationalizability by examining the value functions, which are induced by a bidding strategy. The existence of value-rationalizable strategies with infinite large induced functions is established. We argue that an improper specification of the value function should be attributed to bounded rationality. We show that, under assumptions, strategies which are not value-rationalizable are sub-optimal responses. Finally, the degree of irrationality is assessed by measuring the deviation of the induced function from being a proper value function in terms of its sign and monotonicity. The findings are illustrated by different examples in the independent private value paradigm and the interdependent value setting.

Introduction

Game theory prescriptively suggests Nash equilibrium as a reasonable strategic behavior. Numerous experimental studies, however, have confirmed that bidders tend to deviate from Nash equilibrium points [1,2,3,4]. To address this empirical evidence, a plethora of models depicting non-equilibrium bidding have appeared, such as cursed equilibrium [5], level-k thinking [6], and regret theory [7, 8].

Under common rationality, each player should consistently choose actions which optimize the respective payoff assuming that all agents act similarly. A strategy is rationalizable if it is a best response under the belief that opponents behave in the same manner [9, 10]. Therefore, every player best responds to profiles that are themselves best responses, in an infinite succession of conjectures about the respective competition [11]. In this way, a rationalizable strategy can be “justified” through the common belief that all players act optimally. A stronger notion of rationalizability which assumes point-beliefs is provided by Bernheim [12] and in a different formulation in [13]. In an environment of incomplete preferences, rationalizability is examined in [14, 15]. In the context of a first-price auction with independent private values, an upper bound for rationalizable bidding strategies is studied in [16] and a similar lower bound for discrete bids is derived in [17].

We consider two standard auction formats, a first-price auction and a second-price auction, which are widely used in practice and have been extensively studied in auction theory [18, 19]. We aim to characterize a bidding decision by examining the value function, which allows us to view the strategy as being realized at a Nash equilibrium. Intuitively, an agent can “justify” the adoption of a particular strategy by the belief that the value function assumes a particular form. If the value function, which is implicitly assigned in this manner, is reasonable in the sense that it corresponds to a proper specification, then the strategy will be characterized as value-rationalizable.

To isolate the impact of value-rationalizability, we assume that the underlying distribution of types is common knowledge so that players share a commonly accepted belief about the competition that they face. Consequently, all auction participants make identical probabilistic predictions so that bidders are ex-ante symmetric and share homogeneous beliefs about the competition. In our environment, functional misspecification of the value function becomes the source of irrationality in bidding behavior. Therefore, our contribution is that we extend the notion of rationalizability in the decision-making process capturing a new component of rational thinking, which is related to the form of the value function.

A different identification of the value function is employed in the structural econometric approach of auction theory, which seeks to recover the unobserved values from the observed bids [20,20,22]. Assuming that the bid prices are at the equilibrium point, the value of each bidder can be estimated by the respective bid and a non-parametric estimate of the distribution function and density of the bid prices. In our environment, the underlying signal distribution is common knowledge and, for this reason, the identification of the value function is not based on the distribution of bids. Moreover, we do not aim to the estimation of the values but to a complete specification of the induced value function in order to assess the appropriateness of its form and characterize the rationality of the bid strategy.

The rest of the paper is structured as follows. Section 2 presents the underlying environment in the context of a first-price or a second-price auction. The concept of value-rationalizability is introduced and studied in Section 3 by examining the induced value function for a bidding strategy. The connection between value-rationalizability and irrationality and a proposal for a measure to assess deviation from rationality are investigated in Section 4. Section 5 illustrates the findings with a number of examples. Section 6 concludes.

Model

We assume that n bidders compete against each other for the purchase of an indivisible item. The rules of the auction are well understood by all participants. All players are risk-neutral, so that they maximize their expected monetary profit. Each bidder receives a signal which is a realization of a random variable Xi, i = 1,…,n. The joint probability distribution of the signals FX(x1,…,xn) is absolutely continuous, symmetric in its arguments and it is common knowledge to all players. The support of FX(x1,…,xn) is [0,ω]n. Furthermore, we assume that the signals are affiliated. Suppose that Y = max {Xi, i = 2,…,n} and denote by FY(y|X1) and fY(y|X1) the conditional cumulative distribution function and the density of Y. Then, due to affiliation, the reverse hazard rate fY(y|X1)/FY(y|X1) is increasing in X1.

In general, the exact value of the auctioned object Vi, i = 1,…,n, remains unknown to each bidder. For bidder i, the value of the auctioned object is given by

$$ {v}_i\left(\boldsymbol{x}\right)=E\ \left[{V}_i|\boldsymbol{X}=\boldsymbol{x}\right]=u\ \left({\mathrm{X}}_i,{\mathbf{X}}_{-i}\right) $$

where X = (X1, X2,…, Xn) is the vector of all the signals and Xi denotes the vector (Xj; j ≠ i). The function u (Xi, Xi) is symmetric in its last n−1 arguments. Moreover, writing Yi = max {Xj; j ≠ i}, due to symmetry u (Xi, Xi) = E [Vi|Xi = x,Yi = y] = v(x,y). We denote by bi the bid price of bidder i.

In a first-price auction (FPA), the expected payoff of bidder i, who has received signal Xi = x, is given by

$$ \pi \left({b}_i\right)=E\left[\left({V}_i-{b}_i\right)\kern0.5em \mathbf{1}\left[{b}_i>{b}_i\ast \right]|{X}_i\right] $$

where bi* = max {bj, j ≠ i}. A Nash equilibrium strategy bi maximizes the expected payoff π (bi). When the equilibrium strategy is monotone increasing in the signal, the inverse function of bi* exists and the expected payoff, at the equilibrium point, can be expressed in terms of the maximum signal of the opponents Yi in the form:

$$ \pi \left({b}_i\right)=E\left\{\left(E\left[{V}_i|{X}_i=x,{Y}_i\right]-{b}_i\right)1\left[{\left({b}_i\ast \right)}^{-1}\left({b}_i\right)>{Y}_i\right]|{X}_i=x\right\} $$

where the expectation is with respect to Yi. In the symmetric first-price auction with interdependent values, Milgrom and Weber [23] derived the symmetric pure equilibrium strategy

$$ {\mathrm{B}}_1\left(x,w\right)=\underset{0}{\overset{x}{\int }}w(y) dL\left(y|x\right) $$
(1)

where w(y) = v(y,y) and \( L\left(y|x\right)=\exp \left(\underset{y}{\overset{x}{\int }}\frac{f_Y\left(u|u\right)}{F_Y\left(u|u\right)} du\right) \). The function L(y|x) can be viewed as a cumulative distribution function in y with support [0,x] [19]. Further, r(x) = fY(x|x)/FY(x|x) denotes the reverse hazard rate of FY(x|x) and \( l\left(\left.y\right|x\right)=\frac{dL}{dy}\left(y|x\right) \) is the density of L(y|x).

In a second-price auction (SPA), the payment of bidder i, in case of winning, is bi*, and, therefore, the expected payoff of bidder i, who received signal Xi = x, is

$$ \varPi \left({b}_i\right)=E\left[\left({V}_i-{b}_i\ast \right)\kern0.5em \mathbf{1}\left[{b}_i>{b}_i\ast \right]|{X}_i=x\right] $$

Bidders at the Nash equilibrium point maximize their respective expected profit Π (bi). When the bid prices are monotone in the signal, the inverse function of bi* can be defined. In this case, we may express the expected profit in terms of the maximum signal of the opponents Yi

$$ \varPi \left({b}_i\right)=E\left\{\left(E\left[{V}_i|{X}_i=x,{Y}_i\right]-{b}_i\ast \right)1\left[{\left({b}_i\ast \right)}^{-1}\left({b}_i\right)>{Y}_i\right]|{X}_i=x\right\} $$

with the expectation taken with respect to Yi. Milgrom and Weber [23] derived the symmetric pure equilibrium strategy

$$ {B}_2\left(x,w\right)=w(x) $$
(2)

and Levin and Harstaad [14] showed that this is the only symmetric equilibrium in the SPA.

Value-Rationalizability

Suppose that a bidder, who for convenience and without loss of generality is assumed to be bidder 1, adopts bidding strategy β(x), after having received the signal X1 = x. In the FPA, if we assume that the bidding strategy is at the equilibrium point of Eq. 1, β(x) induces an implicit functional specification for the value function.

Definition 1 In the FPA, a function w(y) is an induced value function of a strategy β(x) if, for each x in [0,ω],

$$ \beta (x)={B}_1\left(x,w\right)=\underset{0}{\overset{x}{\int }}w(y) dL\left(y|x\right)=\underset{0}{\overset{x}{\int }}w(y)l\left(y|x\right) dy $$
(3)

The function w(y) is not uniquely defined. Indeed, when w(y) is modified in a set of measure 0 with respect to the probability distribution L(y|x), the value of the integral remains the same. Therefore, the function w(y) can be defined almost everywhere in the interval [0,ω] with respect to the Lebesgue measure, in the sense that if w1(y) and w2(y) are both induced functions, then w1(y) = w2(y), almost everywhere in [0,ω].

It is also of interest to observe that we cannot fully recover through Eq. 3 the complete underlying value function u(x) or, even, the value function v(x,y). A given bidding strategy characterizes only the value when the maximum signal of the opponents y coincides with the observed signal x, so that the induced value which we can ultimately recover is w(y) = v(y,y).

In the second-price auction environment, we define similarly,

Definition 2 In the SPA, a function w(y) is an induced value function of a strategy β(x) if, for each x in [0,ω],

$$ \beta (x)={B}_2\left(x,w\right)=w(x) $$
(4)

If the induced value w(x) of a bidding strategy β(x) is non-negative and increasing in the signal x, then β(x) can be rationalized as being a symmetric equilibrium.

Definition 3 A strategy β(x) is value-rationalizable if there is an induced value function w(x), which is non-negative and increasing for almost all x in [0,ω].

Value-rationalizability allows a strategy to be viewed as being implemented at the symmetric Nash equilibrium point by its induced value function. Since all players are assumed to be symmetric in the distribution of their signals, the symmetric Nash equilibrium point becomes the best response strategy. Therefore, a value-rationalizable bidding strategy can be interpreted as a rational pricing decision within the game theoretic auction framework under a proper specification of the value function. As a result, value-rationalizable strategies are rational best responses. In the SPA, the derivation of the induced function from the bid price is immediate, since w(x) = β(x). In the FPA, however, the specification of the induced value function is more involved and, for this reason, we proceed to examine it in detail.

In the first-price auction setting, since the exact numerical value of w(0) does not affect the integral in Eq. 3, we may assume, without loss of generality, that w(0) = 0. The symmetric Nash equilibrium point given by Eq. 1 is a monotone increasing strategy. For this reason, we would naturally expect that it will not be possible to view a strategy which is not monotone increasing as being implemented at the symmetric equilibrium. This intuition is made explicit in the following result.

Proposition 1 Suppose that a strategy β(x) is not everywhere increasing. Then, β(x) is not value-rationalizable.

Proof Suppose that for x1 < x2 we have β(x1) > β(x2). From (Eq. 3), \( \underset{x_1}{\overset{x_2}{\int }}w(y)l\left(y|x\right) dy<0 \). Since l(y|x) ≥ 0, we conclude that w(x) < 0 for x in an interval I within [x1, x2]. It follows that β(x) is not value-rationalizable. □

The induced value function of increasing bidding strategies can explicitly be derived.

Proposition 2 Suppose that fY(x|x) > 0 for every x in [0,ω]. Then, the induced value function of an increasing strategy β(x) is given almost everywhere in [0,ω] by

$$ w(x)=\beta (x)+\frac{\beta^{\prime }(x)}{r(x)} $$
(5)

Proof We observe first, that, since β(x) is an increasing function in x, it has a finite derivative almost everywhere on [0,ω] due to [18]. We take x to be in the set in which β(x) is differentiable with a finite derivative. From Eq. 3, differentiation by Leibnitz integral rule yields

$$ {\beta}^{\prime }(x)=w(x)l\left(x|x\right)+\underset{0}{\overset{x}{\int }}w(y)\frac{dl}{dx}\left(y|x\right) dy $$

We note that

$$ l\left(y|x\right)=\frac{dL}{dy}\left(y|x\right)=\exp \left(\hbox{-} \underset{y}{\overset{x}{\int }}\frac{f_Y\left(u|u\right)}{F_Y\left(u|u\right)} du\right)\frac{f_Y\left(y|y\right)}{F_Y\left(y|y\right)}=L\left(y|x\right)\frac{f_Y\left(y|y\right)}{F_Y\left(y|y\right)} $$

and

$$ \frac{dl}{dx}\left(y|x\right)=L\left(y|x\right)\frac{f_Y\left(y|y\right)}{F_Y\left(y|y\right)}\frac{f_Y\left(x|x\right)}{F_Y\left(x|x\right)}=-l\left(y|x\right)\frac{f_Y\left(x|x\right)}{F_Y\left(x|x\right)} $$

Therefore,

$$ {\beta}^{\prime }(x)=w(x)l\left(x|x\right)-\frac{f_Y\left(x|x\right)}{F_Y\left(x|x\right)}\underset{0}{\overset{x}{\int }}w(y)l\left(y|x\right) dy=w(x)l\left(x|x\right)-\mathrm{r}(x)\beta (x) $$

It follows that

$$ w(x)=\frac{\beta^{\prime }(x)+r(x)\beta (x)}{l\left(x|x\right)} $$

Finally, r(x) = l(x|x) implies Eq. 5. □

Corollary 1 Under the assumptions of Proposition 2, w(x) ≥ 0 for all x in [0,ω].

Proof We note that from the assumptions of Proposition 2, fY(x|x) > 0. Further, FY(x|x) ≥ 0 and β(x) is non-negative and increasing, which implies that β΄(x) ≥ 0. We conclude that w(x) ≥ 0 for all x in [0,ω]. □

From Proposition 1, strategies which are not everywhere increasing are not value-rationalizable. Nevertheless, when a bidding strategy β(x) is increasing, its monotonicity does not ensure that it is value-rationalizable. The reason is that its induced value function w(x) does not necessarily have to be increasing in x as required by Definition 3. Examining Eq. 5 more closely, we observe that on one hand β(x) is assumed to be increasing while at the same time r(x) = fY(y|x)/FY(y|x) is non-decreasing due to affiliation. Still, the behavior of the term β΄(x)/r(x) cannot be fully determined unless further assumptions are imposed.

Proposition 3 Suppose that fY(x|x) is differentiable and fY(x|x) > 0 for every x in [0,ω]. Consider a twice differentiable increasing strategy β(x) for which

$$ {\beta}^{\prime }(x)\left(2r(x)-s(x)\right)+{\beta}^{\prime \prime }(x)\ge 0 $$
(6)

where s(x) = fY΄(x|x)/fY(x|x). Then, β(x) is value-rationalizable.

Proof In view of Proposition 1, we need to show that w΄(x) ≥ 0. We note

$$ {w}^{\prime }(x)={\beta}^{\prime }(x)+\frac{\beta^{\prime \prime }(x){F}_Y\left(x|x\right){f}_Y\left(x|x\right)+{\beta}^{\prime }(x){f}_Y{\left(x|x\right)}^2-{\beta}^{\prime }(x){F}_Y\left(x|x\right){f_Y}^{\prime}\left(x|x\right)}{f_Y{\left(x|x\right)}^2} $$

Equivalently,

$$ {w}^{\prime }(x)={\beta}^{\prime }(x)\left(2-\frac{{f_Y}^{\prime}\left(x|x\right)}{f_Y\left(x|x\right)}\frac{1}{r(x)}\right)+\frac{\beta^{\prime \prime }(x)}{r(x)} $$

Since r(x) > 0, Eq. 6 implies that w'(x) ≥ 0. □

Proposition 3 implies the following results.

Corollary 2 Suppose that fY(x|x) > 0, for every x in [0,ω], and r(x) ≥ 2 s(x). Then, a strategy β(x), which is increasing, convex and twice differentiable, is value-rationalizable.

Corollary 3 The induced value function has the property that

$$ \beta (x)=\underset{0}{\overset{x}{\int }}w(y)\frac{f_Y\left(y|y\right)}{F_Y\left(x|x\right)} dy. $$

Proof The result is an immediate consequence of Eq. 6, because

$$ w(x)=\frac{\beta (x){f}_Y\left(x|x\right)+{\beta}^{\prime }(x){F}_Y\left(x|x\right)}{f_Y\left(x|x\right)}=\frac{{\left(\beta (x){F}_Y\left(x|x\right)\right)}^{\prime }}{f_Y\left(x|x\right)}. $$

Irrationality and Value-Rationalizability

Under a symmetric and commonly accepted probabilistic framework, we will show the existence of strategies which are value-rationalizable through arbitrarily large value functions. For a rational strategy, the value of an auctioned object is necessarily bounded in monetary terms. Therefore, we aim to establish the existence of value-rationalizable strategies at Nash equilibrium points which are irrational in nature.

To proceed, we expand the space of induced functions w(x), which we consider through Eq. 1, and assume that the induced value function is almost everywhere continuous in the interval [0,ω]. By definition, the induced value function of a value-rationalizable strategy is increasing in [0,ω]. Since the number of discontinuities of a monotonous function is countable, the induced value function of a value-rationalizable strategy is almost everywhere continuous [18]. We denote by N the set of natural numbers {1,2,3,…}. We can now demonstrate the existence of strategies for which their induced value becomes infinite large.

Proposition 4 In every first-price auction with a symmetric continuous joint probability distribution of signals and for every n∈ N, there is a continuous strategy βn(x) for which the induced value function wn(x) is such that |wn(x)| > n for some interval In of [0,ω].

Proof Consider the space C[0,ω] of all continuous functions on [0,ω] with respect to the supremum norm. Next, define the Volterra operator φ from C[0,ω] to C[0,ω] which maps a continuous function w(x) to

$$ \varphi (w)(x)=\beta (x)=\underset{0}{\overset{x}{\int }}w(y)l\left(y|x\right) dy $$

The Volterra operator φ is completely continuous [24], with ||φ|| = 1. From Proposition 1, the operator φ is invertible. Since C[0,ω] is infinite dimensional, φ does not have a bounded inverse. Therefore, for every n∈ N, there is a strategy βn(x) for which

$$ {\varphi}^{-1}\left({\beta}_{\mathrm{n}}\right)={w}_{\mathrm{n}}\ \mathrm{and}\ \sup \left\{|{w}_{\mathrm{n}}(x)|;x\in \left[0,\omega \right]\right\}>n. $$

Due to the continuity of wn(x) in the compact interval [0,ω], there is a point xn [0,ω], at which sup{|wn(x)|; x ∈ [0,ω]} = |wn(xn)|. We conclude that for bidding strategy βn, the induced value is |wn(xn)| > n. Finally, the continuity of wn(x) yields that for all x in a neighborhood In of xn, |wn(x)| > n. □

The bidding strategy βn(x) of Proposition 4 can be justified as a realization at the symmetric equilibrium of Eq. 1 under the specification of an unbounded induced value function. For an auctioned object of bounded monetary value, a bidder, who implicitly assigns an infinite large value, takes an irrational perspective. In this sense, bidding strategy βn(x) should be considered irrational.

The relationship between irrationality and value-rationalizability will be further explored by addressing the question of whether it is possible for a non-value-rationalizable strategy to be viewed as a best response.

Proposition 5

  1. 1)

    Suppose that in a first-price auction with interdependent values and affiliated signals generated by a commonly known symmetric probability distribution, a strategy β(x) is not value-rationalizable. Then, β(x) is not a best response strategy among the set of pure increasing bidding strategies.

  2. 2)

    Suppose that in a first-price auction with private values which are either independent or symmetric, and the underlying distribution f(x) is commonly known, symmetric and continuously differentiable, a strategy β(x) is not value-rationalizable. Then, β(x) is not a best response strategy among the set of all bidding strategies.

  3. 3)

    In a second-price auction with interdependent values and affiliated signals generated by a commonly known symmetric probability distribution, with n > 2, it is assumed that

$$ E\left[{V}_i|{X}_1={x}_1,{X}_2\le {x}_2,{x_i}^{(1)}\le {X}_i\le {x_i}^{(2)}, fori=3,\dots, n\right]>E\left[{V}_i|{X}_1\le {x}_1,{X}_2={x}_2,{x_i}^{(1)}\le {X}_i\le {x_i}^{(2)}, fori=3,\dots, n\right] $$

for all x1 < x2 and all values xi(j), i = 3,,n, j = 1,2. If β(x) is not value-rationalizable, then β(x) is not a best response strategy among the set of pure continuous increasing bidding strategies.

Proof

  1. 1)

    Suppose that bidder 1 adopts the non-value-rationalizable strategy β(x). We consider any non-negative and almost everywhere increasing value function w(y). Further, we examine the case where all the opponents of bidder 1 choose a monotone increasing bidding strategy under the value function w(y). In the symmetric environment, the unique Nash equilibrium B1(x,w) among pure strategies which are monotone increasing, is given by Eq. 1 due to [25]. Consequently, all the opponents of bidder 1 will adopt the unique Nash equilibrium point provided by Eq. 1. On the other hand, since β(x) is not value-rationalizable, it either fails to take the form of (1), or \( \beta (x)={B}_1\left(x,\tilde{w}\right) \) for an induced value function \( \tilde{w}(y) \), which is not almost everywhere increasing or non-negative. We conclude that β(x) deviates from the unique Nash equilibrium point B1(x,w), which is adopted by all the other agents, and, therefore, it is not the best response strategy among the set of pure increasing bidding strategies.

  2. 2)

    The argument in the second part is similar. In a first-price auction with private values which are either independent or symmetric and the underlying density of the probability distribution is continuously differentiable, the symmetric equilibrium B1 is unique [26]. Therefore, the opponents of bidder 1 are expected to implement this unique Nash equilibrium. As a result, the bidder who follows the non-value-rationalizable strategy β(x) necessarily deviates from the Nash equilibrium point, since β(x) cannot be represented by Eq. 1 for an appropriate non-decreasing and non-negative value function. In conclusion, β(x) is not the best-response strategy to the Nash equilibrium adopted by all other players.

  3. 3)

    In a second-price auction with interdependent values under affiliation and symmetry of the underlying distribution of signal with n > 2, Bikhchandani and Riley [9] showed that when

$$ \mathrm{E}\ \left[{V}_i\ |{X}_1={x}_1,{X}_2\le {x}_2,{x_i}^{(1)}\le {X}_i\le {x_i}^{(2)}, for\ i= 3,\dots, n\right]>\mathrm{E}\ \left[{V}_i\ |{X}_1\le {x}_1,{X}_2={x}_2,{x_i}^{(1)}\le {X}_i\le {x_i}^{(2)}, for\ i= 3,\dots, n\right] $$

for all x1 < x2 and all values xi(j), i = 3,…,n, j = 1,2, the symmetric equilibrium B2 is the unique Nash equilibrium point among pure continuous increasing strategies. If β(x) is not value-rationalizable, then β(x) cannot be represented by Eq. 2 for an appropriate non-decreasing and non-negative value function and deviates from the unique Nash equilibrium. Therefore, β(x) is not the best-response strategy among the set of pure continuous increasing bidding strategies. □

In the setting of Proposition 5, a bidding strategy, which is not value-rationalizable, corresponds to a sub-optimal decision. Failure to adopt a value-rationalizable strategy becomes a manifestation of bounded rationality. In qualitative terms, the implicit use of an induced value function, which is either negative in an interval within [0,ω], or fails to be everywhere increasing, practically means that the bidder assigns either a negative value to some signals or a lower value to a higher signal. Therefore, we may assert that a bidding strategy which is non-value-rationalizable implies that the decision maker behaves in an irrational manner.

In the preceding analysis, we identified the sources of irrationality in relation to the form of the induced value function. In particular, deviations in the non-negative nature (sign) and the non-increasing behavior (monotonicity) of the induced value function generated irrational bidding strategies. Therefore, by capturing the extent of these deviations, we can construct a measure of the departure of the bidding strategy from the rational decision-making framework.

Suppose that the induced value function w(x) for a strategy β(x) has bounded variation, its Jordan decomposition [27] allows us to write w(x) = φ(x) − ψ(x), where φ(x) and ψ(x) are increasing functions and the total variation of w(x) can be expressed as φ(ω) + ψ(ω) − φ(0) − ψ(0). We define the set in which the induced value function is negative and write A = {x: w(x) < 0 for 0 ≤ xω}. The indicator function of A is denoted by 1A(y).

Definition 4 For a strategy β(x) with induced value function w(x), the measure of deviation from rationality R(β) is defined as follows:

  • If w(x) has bounded variation, \( \mathrm{R}\left(\beta \right)=\psi \left(\omega \right)-\psi (0)+\underset{0}{\overset{\omega }{\int }}{1}_A(x){dF}_X(x) \)

  • If w(x) is a function of unbounded variation, R(β) = ∞.

When the induced function has bounded variation, the measure of deviation R(β) has two components. The first one is the total variation of ψ, which precisely corresponds to the non-increasing part in the Jordan decomposition of the value function w(x). In this sense, the total variation of ψ relates to the deviation of w(x) from the required monotonicity. The second component of R(β) captures the probability that w(x) takes negative values, under the probability distribution FX(x). Therefore, the size of R(β) reflects the extent of departure of strategy β(x) from value-rationalizability, and in view of Proposition 5, it can serve as a measure of deviation from rationality.

Proposition 6: A strategy β(x) is value-rationalizable strategy if and only if R(β) = 0.

Proof When R(β) = 0, we have that w(x) = φ(x) and A has measure 0 with respect to F(x). It follows that the induced value function is increasing and non-negative almost everywhere. Conversely, suppose that β(x) is value-rationalizable. Then, w(x) = φ(x), which implies that ψ(x) = 0 for all x. In particular, ψ(ω) − ψ(0) = 0. Further, the function w(x) is almost everywhere non-negative, and hence, the set A has zero probability of occurrence under FX(x). We conclude that, for all x, \( \underset{0}{\overset{\omega }{\int }}{1}_A(x) dFX(x)=0 \). It follows that R(β) = 0. □

Proposition 6 establishes that if R(β) > 0, the strategy β(x) cannot be value-rationalizable. Consequently, under the conditions of Proposition 6, it is not possible to view strategy β(x) as the best response under any specification of the value function, which implies that the bidding behavior under β(x) is irrational.

Examples

We illustrate the concept of value-rationalizability with various examples.

Example 1 In the FPA, suppose that the signals are independent and the underlying probability distribution of the signals is uniform in [0,1] so that FX(x) = x. Then, r(x) = (n−1)/x and L(y|x) = yn−1/xn−1. Further, l(y|x) = (n−1) yn−2/xn−1 and l(x|x) = (n−1)/x = r(x) = fY(x|x). We consider the linear strategy β(x) = ax. Then, from Proposition 2,

$$ w(x)=\frac{a+\left(\left(n-1\right)/x\right) ax}{\left(n-1\right)/x}=a\frac{n}{n-1}x $$
(7)

We easily verify that

$$ \underset{0}{\overset{x}{\int }}w(y) dL\left(y|x\right)=\underset{0}{\overset{x}{\int }} an{\left(n-1\right)}^{-1}y\frac{\left(n-1\right){y}^{n-2}}{x^{n-1}} dy=a\kern0.5em n\frac{x^n}{n{x}^{n-1}}= ax=\beta (x) $$

Therefore, when a > 0, the strategy β(x) is value-rationalizable. For a = 1, β(x) = x is the signal reporting strategy and w(x) = n x/(n−1). For a = (n−1)/n, the strategy becomes the equilibrium in the IPV paradigm with uniform underlying distribution. We also easily confirm from Eq. 7 that, in this case, the induced value function is indeed w(x) = x.

Example 2 We examine a similar environment to the one in Example 1, with independent signals each drawn according to the probability distribution FX(x) = (ex−1)/(e−1), with x in [0,1]. Consequently, r(x) = (n−1)ex (ex−1)−1. For the linear bidding strategy β(x) = α x, Proposition 2 provides,

$$ w(x)=\alpha x+\alpha {\left(n-1\right)}^{-1}\left(1-{\mathrm{e}}^{-x}\right) $$

We note that, for x in [0,1] and α > 0, w΄(x) = α(n−1)−1 [(n−1) + ex)] > 0, and, consequently, the linear bidding strategy with a > 0 is value-rationalizable.

Example 3 We consider a first-price auction with two bidders who receive signals X1 = Z1 + T, X2 = Z2 + T, where Z1, Z2, and T are independent random variables that follow the uniform distribution in [0,1]. In this setting, r(x) = 2/x and L(y|x) = y2x−2. For the linear bidding strategy β(x) = ax, Proposition 2 yields the induced value function

$$ w(x)=a\kern0.5em x+a\frac{x}{2}=\frac{3a\kern0.5em x}{2} $$

Since α > 0, w(x) is increasing in x, and, therefore, the strategy β(x) is value-rationalizable. For α = 2/3, the induced value function becomes w(x) = x. This form is consistent with the value function v(x1,x2) = 0.5(x1 + x2) which produces the symmetric equilibrium strategy \( b(x)=\frac{2}{3}x \) as provided in [19].

Example 4 Returning to the setting of Example 1, with n = 2, we have r(x) = 1/x. We consider the bidding strategy

$$ \beta (x)=\Big\{{\displaystyle \begin{array}{c}x/8\kern1em when\kern0.5em x\le 1/8\\ {}-{x}^2+x/4\kern1em when\kern0.5em x>1/8\end{array}}\operatorname{} $$

From Proposition 2, when x ≤ 1/8, w(x) = x/4. For x > 1/8, w(x) = 0.5x − 3x2.

We observe that for x > 1/6, w(x) < 0 and w΄(x) < 0 when x > 1/8. Consequently, the strategy β(x) is not value-rationalizable, because the induced function was negative for x > 1/6 and decreasing for x > 1/8. The Jordan decomposition of the induced value function w(x) is

$$ \varphi (x)=\left\{{\displaystyle \begin{array}{cc}x/4-1/32& when\kern0.5em x\le 1/80\\ {}0& when\kern0.5em x>1/8\end{array}}\operatorname{}\kern0.5em \mathrm{and}\kern0.5em \psi (x)=\right\{{\displaystyle \begin{array}{cc}-1/{3}^2& when\kern0.5em x\le 1/8\\ {}-0.5x+3{x}^2& when\kern0.5em x>1/8\end{array}}\operatorname{} $$

Consequently, \( \psi (1)-\psi (0)=2.5+\frac{1}{32} \). Furthermore, A = [1/6,1], and, therefore,

$$ \underset{0}{\overset{1}{\int }}{1}_A(x){dF}_X(x)=\underset{1/6}{\overset{1}{\int }} dx=\frac{5}{6} $$

We conclude that \( R\left(\beta \right)=2\frac{7}{48} \), which implies, in view of from Proposition 6, that β(x) should be considered irrational.

Example 5 We examine a bidder, who always places the same bid irrespectively of the signal received, so that β(x) = a, for every x in [0,ω]. In view of Proposition 2, in both the FPA and SPA the induced value function is w(x) = a, for all x in [0,ω]. Therefore, in this case, the value of the auctioned object is perceived to be constant and unrelated to the signal.

Example 6 In a second-price auction, the signals of n > 3 bidders are independent and follow a uniform distribution in [0,1]. We consider the strategy β(x) = xxo, with xo in [0,1]. The induced value function is w(x) = xxo, which is strictly increasing in x. Consequently, ψ(x) = 0 and A = {x: w(x) = 0} = [0,xo]. The measure of deviation from irrationality for this strategy is \( R\left(\beta \right)=\underset{0}{\overset{1}{\int }}{1}_A(x)d{F}_X(x)=\underset{0}{\overset{x_o}{\int }} dx={x}_o \). Therefore, the irrationality of strategy β(x) is characterized by the size of xo.

Conclusion

The concept of value-rationalizability allows us to visualize irrationality as a violation of a proper assignment of the value function. In the preceding analysis, for a given strategy, we examined the induced value function in order to assess its properties in terms of sign and monotonicity. For a first-price and a second-price auction, we have shown that, under assumptions, strategies which are not value-rationalizable are sub-optimal responses to the competition, and, in this sense, they may be characterized as irrational. Moreover, the extent of deviation of the induced value function with respect to the correct sign (non-negative) and monotonicity (increasing) has been built into a measure of irrationality. This measure becomes useful in ranking bidding strategies, which are driven by bounded rationality.

In the environment, which we have considered, all players symmetrically share the same belief about the underlying probabilistic framework, so that deviations from best-response actions can be attributed only to inappropriate specification of the corresponding value function. Moving to a more general setting, agents may also act under different beliefs about the probability distribution of the signals, which would “justify” their pricing decisions. Therefore, in principle, the concept of value-rationalizability can be generalized, accommodating different beliefs about the competition.

Our results have been limited to the mechanism of a first-price auction and second-price auction with interdependent values. Another challenging direction for future research would be to investigate other types of auction mechanisms, which may also incorporate reselling possibilities, budget constraints, sequential multi-unit sales, and multi-dimensional types.

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Lorentziadis, P.L. Value-Rationalizability in Auction Bidding. SN Oper. Res. Forum 1, 12 (2020). https://doi.org/10.1007/s43069-020-0012-y

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Keywords

  • Bidding
  • Auctions
  • Rationalizability
  • Induced value function
  • Irrationality