Abstract
In a symmetric firstprice or secondprice auction, a bidding strategy is characterized as valuerationalizable if it can be viewed as being realized at a Nash equilibrium under the specification of a nonnegative and increasing value function. In an environment where the underlying probabilistic framework is common knowledge, we investigate conditions for valuerationalizability by examining the value functions, which are induced by a bidding strategy. The existence of valuerationalizable 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 valuerationalizable are suboptimal 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 nonequilibrium bidding have appeared, such as cursed equilibrium [5], levelk 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 pointbeliefs 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 firstprice 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 firstprice auction and a secondprice 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 valuerationalizable.
To isolate the impact of valuerationalizability, 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 exante 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 decisionmaking 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 nonparametric 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 firstprice or a secondprice auction. The concept of valuerationalizability is introduced and studied in Section 3 by examining the induced value function for a bidding strategy. The connection between valuerationalizability 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 riskneutral, so that they maximize their expected monetary profit. Each bidder receives a signal which is a realization of a random variable X_{i}, i = 1,…,n. The joint probability distribution of the signals F_{X}(x_{1},…,x_{n}) is absolutely continuous, symmetric in its arguments and it is common knowledge to all players. The support of F_{X}(x_{1},…,x_{n}) is [0,ω]^{n}. Furthermore, we assume that the signals are affiliated. Suppose that Y = max {X_{i}, i = 2,…,n} and denote by F_{Y}(yX_{1}) and f_{Y}(yX_{1}) the conditional cumulative distribution function and the density of Y. Then, due to affiliation, the reverse hazard rate f_{Y}(yX_{1})/F_{Y}(yX_{1}) is increasing in X_{1}.
In general, the exact value of the auctioned object V_{i}, i = 1,…,n, remains unknown to each bidder. For bidder i, the value of the auctioned object is given by
where X = (X_{1}, X_{2},…, X_{n}) is the vector of all the signals and X_{−i} denotes the vector (X_{j}; j ≠ i). The function u (X_{i}, X_{−i}) is symmetric in its last n−1 arguments. Moreover, writing Y_{i} = max {X_{j}; j ≠ i}, due to symmetry u (X_{i}, X_{−i}) = E [V_{i}X_{i} = x,Y_{i} = y] = v(x,y). We denote by b_{i} the bid price of bidder i.
In a firstprice auction (FPA), the expected payoff of bidder i, who has received signal X_{i} = x, is given by
where b_{i}* = max {b_{j}, j ≠ i}. A Nash equilibrium strategy b_{i} maximizes the expected payoff π (b_{i}). When the equilibrium strategy is monotone increasing in the signal, the inverse function of b_{i}* exists and the expected payoff, at the equilibrium point, can be expressed in terms of the maximum signal of the opponents Y_{i} in the form:
where the expectation is with respect to Y_{i}. In the symmetric firstprice auction with interdependent values, Milgrom and Weber [23] derived the symmetric pure equilibrium strategy
where w(y) = v(y,y) and \( L\left(yx\right)=\exp \left(\underset{y}{\overset{x}{\int }}\frac{f_Y\left(uu\right)}{F_Y\left(uu\right)} du\right) \). The function L(yx) can be viewed as a cumulative distribution function in y with support [0,x] [19]. Further, r(x) = f_{Y}(xx)/F_{Y}(xx) denotes the reverse hazard rate of F_{Y}(xx) and \( l\left(\left.y\rightx\right)=\frac{dL}{dy}\left(yx\right) \) is the density of L(yx).
In a secondprice auction (SPA), the payment of bidder i, in case of winning, is b_{i}*, and, therefore, the expected payoff of bidder i, who received signal X_{i} = x, is
Bidders at the Nash equilibrium point maximize their respective expected profit Π (b_{i}). When the bid prices are monotone in the signal, the inverse function of b_{i}* can be defined. In this case, we may express the expected profit in terms of the maximum signal of the opponents Y_{i}
with the expectation taken with respect to Y_{i}. Milgrom and Weber [23] derived the symmetric pure equilibrium strategy
and Levin and Harstaad [14] showed that this is the only symmetric equilibrium in the SPA.
ValueRationalizability
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 X_{1} = 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],
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(yx), 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 w_{1}(y) and w_{2}(y) are both induced functions, then w_{1}(y) = w_{2}(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 secondprice 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],
If the induced value w(x) of a bidding strategy β(x) is nonnegative and increasing in the signal x, then β(x) can be rationalized as being a symmetric equilibrium.
Definition 3 A strategy β(x) is valuerationalizable if there is an induced value function w(x), which is nonnegative and increasing for almost all x in [0,ω].
Valuerationalizability 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 valuerationalizable 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, valuerationalizable 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 firstprice 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 valuerationalizable.
Proof Suppose that for x_{1} < x_{2} we have β(x_{1}) > β(x_{2}). From (Eq. 3), \( \underset{x_1}{\overset{x_2}{\int }}w(y)l\left(yx\right) dy<0 \). Since l(yx) ≥ 0, we conclude that w(x) < 0 for x in an interval I within [x_{1}, x_{2}]. It follows that β(x) is not valuerationalizable. □
The induced value function of increasing bidding strategies can explicitly be derived.
Proposition 2 Suppose that f_{Y}(xx) > 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
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
We note that
and
Therefore,
It follows that
Finally, r(x) = l(xx) 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, f_{Y}(xx) > 0. Further, F_{Y}(xx) ≥ 0 and β(x) is nonnegative 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 valuerationalizable. Nevertheless, when a bidding strategy β(x) is increasing, its monotonicity does not ensure that it is valuerationalizable. 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) = f_{Y}(yx)/F_{Y}(yx) is nondecreasing 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 f_{Y}(xx) is differentiable and f_{Y}(xx) > 0 for every x in [0,ω]. Consider a twice differentiable increasing strategy β(x) for which
where s(x) = f_{Y}΄(xx)/f_{Y}(xx). Then, β(x) is valuerationalizable.
Proof In view of Proposition 1, we need to show that w΄(x) ≥ 0. We note
Equivalently,
Since r(x) > 0, Eq. 6 implies that w'(x) ≥ 0. □
Proposition 3 implies the following results.
Corollary 2 Suppose that f_{Y}(xx) > 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 valuerationalizable.
Corollary 3 The induced value function has the property that
Proof The result is an immediate consequence of Eq. 6, because
Irrationality and ValueRationalizability
Under a symmetric and commonly accepted probabilistic framework, we will show the existence of strategies which are valuerationalizable 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 valuerationalizable 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 valuerationalizable strategy is increasing in [0,ω]. Since the number of discontinuities of a monotonous function is countable, the induced value function of a valuerationalizable 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 firstprice auction with a symmetric continuous joint probability distribution of signals and for every n∈ N, there is a continuous strategy β_{n}(x) for which the induced value function w_{n}(x) is such that w_{n}(x) > n for some interval I_{n} 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
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 β_{n}(x) for which
Due to the continuity of w_{n}(x) in the compact interval [0,ω], there is a point x_{n} ∈ [0,ω], at which sup{w_{n}(x); x ∈ [0,ω]} = w_{n}(x_{n}). We conclude that for bidding strategy β_{n}, the induced value is w_{n}(x_{n}) > n. Finally, the continuity of w_{n}(x) yields that for all x in a neighborhood I_{n} of x_{n}, w_{n}(x) > n. □
The bidding strategy β_{n}(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 β_{n}(x) should be considered irrational.
The relationship between irrationality and valuerationalizability will be further explored by addressing the question of whether it is possible for a nonvaluerationalizable strategy to be viewed as a best response.
Proposition 5

1)
Suppose that in a firstprice auction with interdependent values and affiliated signals generated by a commonly known symmetric probability distribution, a strategy β(x) is not valuerationalizable. Then, β(x) is not a best response strategy among the set of pure increasing bidding strategies.

2)
Suppose that in a firstprice 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 valuerationalizable. Then, β(x) is not a best response strategy among the set of all bidding strategies.

3)
In a secondprice auction with interdependent values and affiliated signals generated by a commonly known symmetric probability distribution, with n > 2, it is assumed that
for all x_{1} < x_{2} and all values x_{i}^{(j)}, i = 3,…,n, j = 1,2. If β(x) is not valuerationalizable, then β(x) is not a best response strategy among the set of pure continuous increasing bidding strategies.
Proof

1)
Suppose that bidder 1 adopts the nonvaluerationalizable strategy β(x). We consider any nonnegative 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 B_{1}(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 valuerationalizable, 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 nonnegative. We conclude that β(x) deviates from the unique Nash equilibrium point B_{1}(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)
The argument in the second part is similar. In a firstprice auction with private values which are either independent or symmetric and the underlying density of the probability distribution is continuously differentiable, the symmetric equilibrium B_{1} 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 nonvaluerationalizable strategy β(x) necessarily deviates from the Nash equilibrium point, since β(x) cannot be represented by Eq. 1 for an appropriate nondecreasing and nonnegative value function. In conclusion, β(x) is not the bestresponse strategy to the Nash equilibrium adopted by all other players.

3)
In a secondprice auction with interdependent values under affiliation and symmetry of the underlying distribution of signal with n > 2, Bikhchandani and Riley [9] showed that when
for all x_{1} < x_{2} and all values x_{i}^{(j)}, i = 3,…,n, j = 1,2, the symmetric equilibrium B_{2} is the unique Nash equilibrium point among pure continuous increasing strategies. If β(x) is not valuerationalizable, then β(x) cannot be represented by Eq. 2 for an appropriate nondecreasing and nonnegative value function and deviates from the unique Nash equilibrium. Therefore, β(x) is not the bestresponse strategy among the set of pure continuous increasing bidding strategies. □
In the setting of Proposition 5, a bidding strategy, which is not valuerationalizable, corresponds to a suboptimal decision. Failure to adopt a valuerationalizable 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 nonvaluerationalizable 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 nonnegative nature (sign) and the nonincreasing 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 decisionmaking 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 ≤ x ≤ ω}. The indicator function of A is denoted by 1_{A}(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 nonincreasing 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 F_{X}(x). Therefore, the size of R(β) reflects the extent of departure of strategy β(x) from valuerationalizability, and in view of Proposition 5, it can serve as a measure of deviation from rationality.
Proposition 6: A strategy β(x) is valuerationalizable 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 nonnegative almost everywhere. Conversely, suppose that β(x) is valuerationalizable. 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 nonnegative, and hence, the set A has zero probability of occurrence under F_{X}(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 valuerationalizable. 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 valuerationalizability 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 F_{X}(x) = x. Then, r(x) = (n−1)/x and L(yx) = y^{n−1}/x^{n−1}. Further, l(yx) = (n−1) y^{n−2}/x^{n−1} and l(xx) = (n−1)/x = r(x) = f_{Y}(xx). We consider the linear strategy β(x) = ax. Then, from Proposition 2,
We easily verify that
Therefore, when a > 0, the strategy β(x) is valuerationalizable. 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 F_{X}(x) = (e^{x}−1)/(e−1), with x in [0,1]. Consequently, r(x) = (n−1)e^{x} (e^{x}−1)^{−1}. For the linear bidding strategy β(x) = α x, Proposition 2 provides,
We note that, for x in [0,1] and α > 0, w΄(x) = α(n−1)^{−1} [(n−1) + e^{−x})] > 0, and, consequently, the linear bidding strategy with a > 0 is valuerationalizable.
Example 3 We consider a firstprice auction with two bidders who receive signals X_{1} = Z_{1} + T, X_{2} = Z_{2} + T, where Z_{1}, Z_{2}, and T are independent random variables that follow the uniform distribution in [0,1]. In this setting, r(x) = 2/x and L(yx) = y^{2}x^{−2}. For the linear bidding strategy β(x) = ax, Proposition 2 yields the induced value function
Since α > 0, w(x) is increasing in x, and, therefore, the strategy β(x) is valuerationalizable. For α = 2/3, the induced value function becomes w(x) = x. This form is consistent with the value function v(x_{1},x_{2}) = 0.5(x_{1} + x_{2}) 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
From Proposition 2, when x ≤ 1/8, w(x) = x/4. For x > 1/8, w(x) = 0.5x − 3x^{2}.
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 valuerationalizable, 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
Consequently, \( \psi (1)\psi (0)=2.5+\frac{1}{32} \). Furthermore, A = [1/6,1], and, therefore,
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 secondprice auction, the signals of n > 3 bidders are independent and follow a uniform distribution in [0,1]. We consider the strategy β(x) = x − x_{o}, with x_{o} in [0,1]. The induced value function is w(x) = x − x_{o}, which is strictly increasing in x. Consequently, ψ(x) = 0 and A = {x: w(x) = 0} = [0,x_{o}]. 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 x_{o}.
Conclusion
The concept of valuerationalizability 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 firstprice and a secondprice auction, we have shown that, under assumptions, strategies which are not valuerationalizable are suboptimal 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 (nonnegative) 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 bestresponse 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 valuerationalizability can be generalized, accommodating different beliefs about the competition.
Our results have been limited to the mechanism of a firstprice auction and secondprice 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 multiunit sales, and multidimensional types.
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Lorentziadis, P.L. ValueRationalizability in Auction Bidding. SN Oper. Res. Forum 1, 12 (2020). https://doi.org/10.1007/s430690200012y
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Keywords
 Bidding
 Auctions
 Rationalizability
 Induced value function
 Irrationality