Information Provision in a Sequential Search Setting
Consider a variation on the classic Weitzman search problem, in which competing firms can choose how much information about their product to reveal to a consumer. If there are no search frictions, there is a unique symmetric equilibrium in pure strategies; and for any market of finite size, the firms are not fully informative. With search frictions, if the expected value of the prize is sufficiently high, there is a unique symmetric equilibrium in which firms are fully informative. Remarkably, a small search cost leads to the perfect competition level of information provision–consumers gain when firms are forced to compete on information.
Keywords: Search; Multiple Senders; Information Transmission.
JEL Classifications: C72; D82; D83
This study explores information provision in a problem of sequential search. In particular, we focus on the classic “Weitzman” search problem, as described in Weitzman (1979) [wei]. An agent faces a number of boxes, each containing some unknown random prize whose distribution is known to the agent. Time is discrete and each period, the agent may investigate or inspect any box that she chooses. She may only investigate one box per time interval, and each box has a (possibly random) search cost as well as a search time associated with the inspection of its contents. Having investigated a box, the agent always may return to choose the prize contained within.
We assume the classic setup, in which an agent must inspect a box before she can claim it.111Perhaps surprisingly, this assumption is non-trivial, as illustrated in the recent paper by Doval (2017) [dov]. Consequently, the agent has two decisions to make: she must decide in which order to sample the boxes, and she must decide when to stop. From [wei], the solution to this problem takes a simple form. Each box is assigned a value, a reservation price characteristic to that box alone, and the agent simply inspects the highest value box first, then the next highest value box, etc. If at any time the agent has already investigated a box with a reward higher than the highest value of the remaining boxes, the agent stops the search and the decision problem is over.
We extend this problem by endowing the boxes themselves with agency. As in the canonical problem, each box has an exogenous, commonly known distribution over its reward, search cost, and search time. However, now the boxes are not forced to fully reveal their contents upon inspection by the agent. Instead, each box commits before knowing its reward to a signal or experiment, the results of which the agent observes during inspection. The agent is a rational Bayesian, and so updates her beliefs about the contents of the box after her visit. Importantly, she observes the experiment (but not the realization) chosen by each box before commencing search.222The problem in [wei] can be seen as the special case where the signal designed by each box is fully revealing. Naturally, each box wants to be selected and chooses a signal that maximizes the probability of it being selected, given an optimal search procedure by the agent.
Consider the following example. A consumer wants to buy a good of uncertain quality and the several sellers carrying it do not compete on price. It is costly for the consumer to visit a seller, but visiting a seller can be helpful: each seller can choose to reveal information about the product to the consumer upon her visit. One particular manifestation of this could be stores that allow the consumer to “try the product” in the store, versus stores that do not: the former would correspond to a more informative experiment than the latter. In particular, this model can be used to (at least partially) explain why car dealerships in the US offer test drives, why primary schools have open houses, or why clothing stores allow people to try clothes on. Alternatively, as pointed out by Au [au3], the scenario may be interpreted as modeling the design of product varieties. Consumers and firms are unsure of consumers’ match values with a product, and a (costly) visit helps consumers discover this. Hence, this scenario poses the question as to how competing firms should design distributions over product designs optimally.
For simplicity, we restrict attention to the scenario in which each of the boxes contains a random prize of either high or low quality, where the probability of high quality, , is the same for each box. Without loss of generality, we normalize the values of the low- and high-quality prize to the risk-neutral searcher as and , so is the expected value of the prize under the prior probability of high quality. Hence, the problem of experiment design for the boxes is equivalent to one where the boxes choose any distribution supported on a subset of such that expected value of the prize under this distribution is .
In the ensuing analysis we distinguish between the frictionless case and the case with frictions. That is, in the general model of Weitzman search, the searcher must pay some cost to inspect a box, and is impatient, discounting the future by some positive . Accordingly, the frictionless case refers to the situation where and .333Alternatively, instead of interpreting the scenario with and as the frictionless case, we could think of this case as one in which the searcher sees all of the signals simultaneously. This interpretation would also lend support to our focus on fair subgame perfect equilibria. Without frictions, a unique fair444Defined formally infra. Essentially, the fair assumption imposes that Pandora randomize uniformly when she is indifferent. symmetric equilibrium in pure strategies exists; and crucially, for any finite the optimal experiment design is not fully informative. However, as the number of boxes becomes infinitely large, the equilibrium converges to one in which each box uses a fully informative signal. This can be thought of as the Perfect Competition level of information provision.
If instead there are frictions–there is some nonzero cost of inspecting a box and/or the searcher is relatively impatient–the equilibrium structure is different. If the mean is sufficiently high, , then the unique symmetric equilibrium is for each box to choose a fully informative signal (Theorem 4.1). There is no profitable deviation from the fully-informative distribution since any deviation would ensure that the deviating box be visited last. Then, since the mean (or equivalently the probability of low quality under the prior) is so high, the probability of the deviating box being visited by the searcher, and hence being selected, is also low.
Furthermore, for any possible value of , there is no symmetric pure strategy equilibrium in which boxes each choose a signal that is not fully informative. There is always a profitable deviation in which a box can be slightly more informative and thereby ensure that it is visited first, earning it a discrete jump in its payoff. However, if the mean is too low then there is no symmetric equilibrium in pure strategies, since there is a profitable deviation from the vector of fully informative signals. In spite of this, we show in Theorem 2.8 that a symmetric mixed strategy equilibrium must exist.
We finish by comparing the searcher’s welfare in the cases with and without search frictions (Theorem 5.1). Remarkably, for any finite number of boxes (and for sufficiently high ), a small cost is actually welfare improving for a searcher and leads to the perfect competition level of information provision. In this model of search, the boxes compete with information, not price and crucially they choose what information the searcher will receive before she starts her search. Because there is a search cost, the choice of the order in which to visit the boxes becomes paramount. Hence, frictions force sellers to compete on the order in which they are searched, and that additional competition benefits the searcher. She receives more information, which makes it more likely that she selects the box with the highest realized quality.
Another way that we can think about these two scenarios is as a comparison of search where the consumer has to visit a seller (at cost) in order to see the signal realization about the product versus the situation where all of the signal realizations arrive simultaneously for free. Think of, for instance, shopping for a product in person in contrast to using a comparison website to shop. The results of this paper suggest that for the consumer, there could be a downside to having the information immediately available at her fingertips. Furthermore, this interpretation lends further support to our stipulation of a fair tie breaking rule. With all of the signal realizations on a screen, it is reasonable to suppose (via some sort of Laplacian “Principle of Insufficient Reason” argument) that the one the consumer will look at first will be random.
1.1 Related Work
The paper closest in content to this paper is “Competitive Information Disclosure in Search Markets” (Board and Lu (2017)) [board]. There, the authors examine the case of a collection of sellers competing to persuade a prospective buyer to purchase one of their products. Each seller has the same set of products to sell and designs an experiment in order to sway the buyer’s opinion on the product. In contrast to our paper, buyers pick a seller at random; and moreover, the state of the world upon which the searchers’ utility depends is common. Here, the information provided by each box provides the searcher with no information about the contents of any of the other boxes.
Another close related paper is “Competition over Agents with Boundedly Rational Expectations” (Spiegler (2006)) [spiegler]. The unique equilibrium in that paper is essentially isomorphic to the specific case in the frictionless model when the mean is fixed at . Boleslavsky and Cotton (2015) [cotton] and Albrecht (2017) [al] each derive results that characterize the the two-player solution to this problem of (frictionless) competitive information provision. Other papers that look at competitive information provision include Koessler, Laclau, and Tomala (2017) [Koessler]; Boleslavsky and Cotton (2016) [bol]; and a pair of papers by Au and Kawai (2017) [Au1, Au2]. In the second of these papers, [Au2], the authors show that in the competitive persuasion setting (equivalent to the frictionless case in this paper), the first order statistic of the other players’ distributions has the linear form implied by our characterization of the equilibrium. They also show that each player’s distribution has no point masses, except possibly on , and that in the limit each player chooses the prior distribution (full information revelation). Garcia (2017) [gar], goes beyond the basic competitive persuasion environment, and looks at the optimal policies for sellers in a frictionless search market who can compete both on information provision and price.
Note that this idea of competitive persuasion concerns a different sort of competition than that featured in Gentzkow and Kamenica (2017, 2017) [comp, rich], or Norman and Li (2017) [li]. In those problems, the persuaders inhabit the same common state and independently choose experiments to convey information about this common state. In contrast, the boxes here each present information about their own i.i.d. draws, and not some common state. In addition, this idea of information provision as the choice of a Blackwell experiment is also used in, among others, Gentzkow and Kamenica (2010) [gent]; Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017) [kol]; and Perez and Skreta (2017) [skreta].
There are a number of papers in the search literature that bear mentioning as well. In particular, the vein of research that focuses on obfuscation is relevant. Two such papers are Ellison and Wolitzky (2009) [wol] and Ellison and Ellison (2009) [ell]. In the first, the authors extend the model of Stahl (1989) [sta] by allowing firms to choose the length of time it takes for consumers to learn its price. Allowing for such delays hurts consumers, since obfuscation leads to longer search times and higher prices. Ellison and Ellison, in turn, provide empirical evidence suggesting that as technology has made price search easier for consumers, firms have responded by taking actions that make price search more difficult.
More recently, Gamp and Krähmer (2017) [gamp] examine a scenario in which sellers can dupe naive consumers into buying products that are lemons. As in this model (though through a different mechanism), search frictions can be beneficial to consumers. Choi, Dai and Kim (2016) [choi2] incorporate price competition into a model of Weitzman search and show, among other things, that a reduction in search cost can lead to an increase in price. Lastly, Perez-Richet (2012) [pr] considers a model in which candidates can choose whether to disclose information in order to convince a decision maker to adopt their product.
After an earlier draft of this paper [perc] was posted on ArXiv, another paper, Au (2018) [au3], that looks at the identical problem of information provision in the context of Weitzman search, was posted as a working paper. There the author looks at competition as an stimulus for dispersion in product designs. Au establishes some further results (beyond those in this paper) concerning properties of the mixed strategy equilibrium, and is able to explicitly characterize a mixed-strategy equilibrium for a sufficiently high outside value. Au also includes novel comparative statics results that center on the boxes’ welfare as affected by the various environmental parameters.
This paper is the first to look at the Weitzman search problem as an information provision problem for the boxes themselves. The simple form of the model allows for tractability and clean results. The main contribution of the paper–that introduction of a slight search cost leads to the perfect competition level of information provision–is novel in the literature.
2 The Model
This game consists of players; one agent, (Pandora), and ex ante identical boxes indexed by . Each box has a random quality to the consumer of either or , and is the probability of the quality being . We call this prior and impose that each boxes quality is independent. In addition, a pair of inspection costs, , is associated with each box. Pandora’s utility function is linear in a box’s quality.
Each box has a compact metric space of signal realizations . Prior to learning its quality, each box chooses a signal, Borel measurable function , and has the power of full commitment to the signal. For any box , each signal realization leads to a posterior distribution with the corresponding posterior mean . Accordingly, the signal leads to a distribution over posterior means, .
We impose that Pandora’s reservation value or outside option is , and that the search costs and are sufficiently low that Pandora finds it optimal to search i.e. . Pandora, before embarking on her search, observes the signal (or experiment) chosen by each of the boxes. Accordingly, she knows how informative each box has decided to be about its contents before she decides whom to visit. When Pandora visits a box, she observes the signal realization, and updates her prior as to the (expected) value of the prize within the box.
Each box’s problem is equivalent to one in which each box chooses a distribution supported on with mean .
It is a standard result in the literature (see e.g. [gent]) that the choice of signal provision by a box is one of choosing a feasible distribution of posterior means. Elton and Hill [hill] define this set of feasible distributions for a given prior as the set of “Fusions” of distribution , . Proposition 3.13 in [hill] illustrates that if is a Bernoulli distribution, then this set, , is the set of all distributions on with mean . Henceforth, we call this set . ∎
We stipulate box ’s utility in the game to be the probability that it is the box selected when Pandora chooses to stop searching and select a box. Accordingly, this game is a constant sum game: the payoffs of all the boxes sum to .
Given a Bernoulli prior with mean , the set of Pure Strategies for each box is the set of distributions supported on with mean . That is, a pure strategy for box consists of a choice of distribution .
The set of mixed strategies is defined as expected:
The set of Mixed Strategies, , is the set of all Borel probability distributions over the pure strategies of box :
A typical element is a Borel probability distribution over the (compact and equipped with a metric) set of box ’s pure strategies.
The timing of the game is
Each box simultaneously chooses a mixed strategy .
For each box, the outcome of the randomization over pure strategies is realized.
Pandora observes each distribution over prizes and chooses an optimal policy, , which consists of a selection rule dictating the order in which she should inspect the boxes, and a stopping rule deciding when she should stop searching.
2.1 Pandora’s Problem
Given the choice of distributions by each of the boxes, Pandora is faced with boxes, each containing an expected reward distributed according to a probability distribution function , which is an endogenous choice of box . The seminal result of Weitzman ([wei]) was to show that each box can be fully characterized by its reservation price:
The reservation price of a box is , where is defined as
Given this definition, Weitzman ([wei]) proved the following proposition:
An optimal (pure strategy) policy, , or “Pandora’s Rule” ([wei]), is completely characterized by the following two rules:
Selection Rule: If a box is to be visited and examined, it should be the closed box with the highest reservation price.
Stopping Rule: Search should be stopped whenever the maximum reservation price of the uninspected boxes is lower than the maximum sampled reward, .
Note that at each point in Pandora’s search we can partition the set of boxes, , into opened and unopened boxes, and , respectively. Hence, is given formally as
Let be the Cartesian product, . Given a vector of mixed strategies by the other boxes and an optimal policy by Pandora, box ’s expected payoff from choosing mixed strategy is
Of course, there is a pressing issue yet to be resolved: there are two natural occasions in Pandora’s search at which she may be indifferent, and we have to determine a rule in order to resolve this indifference. The rule we find the most sensible is one of fairness; that at these points of indifference Pandora randomizes fairly over the boxes responsible for the indifference. In order to express this precisely, we introduce some helpful notation. Define as the set of boxes that induce the maximal value of all of the boxes in :
Additionally, define as the set of boxes whose sampled reward is equal to the maximum sampled reward:
We call a policy, , Fair if
Whenever is not a singleton, Pandora visits each member of next with equal probability; and
Whenever is not a singleton, Pandora selects each member of with equal probability.
Then, in Fair Subgame Perfect Equilibrium, given Pandora’s optimal fair policy, , each box must choose a signal in order to maximize the chance that it is chosen by Pandora, given the optimal signals by the other boxes. Formally,
A mixed strategy Fair Subgame Perfect Equilibrium (FSPE) consists of a policy, , for the searcher and a vector of strategies for the boxes, , such that
Pandora chooses a Fair optimal policy:
’s optimal policy, , satisfies ; and
’s optimal policy is Fair.
For each box , .
Having fully established what Pandora will do upon seeing the signal choices by the boxes, we can work backward and focus our attention on the strategic interaction between the boxes.555Henceforth we use FSPE, SPE, equilibrium, and subgame perfect equilibrium interchangeably–to refer to this idea of FSPE.
Because the game is discontinuous, the existence of equilibria isn’t immediately evident. Results developed in Reny (1999) [reny] allow us to derive the following theorem; though, as we show later on, there is not generally an equilibrium in pure strategies.
A fair subgame perfect equilibrium in mixed strategies always exists.
We present a sketch of the proof here, with further detail left to Appendix A.1. This existence result holds for a general prior, not merely one with binary support.
First, the strategy set for each box is quite well-behaved: the set of pure strategies and the set of mixed strategies are both compact, convex, and Hausdorff. Second, because the game is constant sum–recall that the payoff for each box is simply the probability that it is selected at the end of Pandora’s search–the game is reciprocally upper-semicontinuous. As intuitively described in [reny], “whenever some player’s payoff jumps down, some other player’s payoff jumps up.” Third, the mixed extension of the game is payoff secure. In fact, we show a stronger condition–one that obviously implies payoff security–that each box’s payoff is lower-semicontinuous in the other boxes’ mixed strategies. The properties of reciprocal upper-semicontinuity and payoff security ensure there is a mixed strategy equilibrium. ∎
We wish to remark here that this existence result is relatively independent of our chosen tie-breaking rule (the “fairness” restriction). For instance, we could allow Pandora to “prefer” any of the sellers by visiting them with a probability greater than when indifferent, which would not affect the mechanics of the proof. It is in the frictionless case that the fairness assumption has bite–there it serves to exclude a certain portion of the set of symmetric equilibria.
3 Frictionless Search
We begin with the case where the inspection cost, , is equal to , and the discount rate for the searcher, , is equal to . We call this frictionless search. Alternatively, this can be thought of the situation in which Pandora observes all of the signal realizations simultaneously. Indeed, this interpretation lends itself naturally to the focus on fair equilibria.
If and , then this scenario is equivalent one in which a mean is fixed, and the boxes each choose a random variable distributed on with the given mean. The winner of the game is the player whose random variable has the highest realization.
We have the following proposition (with proof left to Appendix B.1):
(Frictionless) There are no symmetric equilibria with point masses on any point in the interval .
The mechanics behind this result are clear. There can be no mass points on any point other than since a box can always deviate by moving its mass point infinitesimally higher and achieving a discrete jump in its payoff. We can quickly gain some intuition for this result by thinking about the symmetric vector of strategies where each box chooses a fully informative signal. Any box can deviate profitably by instead putting some ( small) weight on and weight on some strictly positive point (very) close to . Since is so small, this deviation will “cost” the deviator next to nothing, but since the probability that all the other boxes all have a realization of is strictly positive, the deviator will have secured itself a discrete jump up in its payoff.
Next, we characterize the unique symmetric equilibrium:
If then the unique symmetric equilibrium is for each box to play , defined as
where and ; and .
If then the unique symmetric subgame perfect equilibrium is for each box to play strategy supported on .
We present a sketch of the proof, with further detail left to Appendix B.2. Showing this formulation constitutes an equilibrium is simple: direct calculation allows us to verify that there is no profitable deviation. Establishment of uniqueness proceeds in two steps. First, we prove the following lemma:
(Frictionless) Suppose that in a symmetric equilibrium each box puts a point mass of size on . Then, must satisfy .
If the mean is sufficiently high, then any symmetric equilibrium must have a point mass on , since otherwise a box could deviate profitably by choosing a fully informative signal. Against boxes choosing strategies without point masses on , a box playing the prior would be selected for sure after the high realization, and so if the high realization is sufficiently likely, the other boxes need to take steps to thwart this.
In the second step, we can solve for the equilibrium strategy directly, using optimal control techniques. We fix a vector of strategies, , for the other boxes: . Given , we define the functional as the Euler-Lagrange equation
The first constraint ensures the distribution satisfies Kolmogorov’s second axiom, and the second constraint guarantees that the expectation is . Box simply solves this maximization problem. We finish by generating some smaller auxiliary results that are needed in order to pin down the support of . ∎
As evinced by our next result, competition has a clear effect on information provision in the frictionless case.
(Frictionless) Let . If the number of boxes is increased, the weight placed on in the symmetric equilibrium must increase. That is, is strictly increasing in .
In the limit, as the number of boxes, , becomes infinitely large, the weight on converges to . That is, the equilibrium distribution converges to a distribution with support on two points, and .
For convenience, define , and recall (see Theorem 3.2) that we have
Define the right hand side of this expression as . For , is decreasing in and therefore increasing in over the same interval. Moreover, we make take the partial derivative with respect to :
Thus, as increases, the needed to satisfy the above expression must increase. That is, more and more weight is put on . Concurrently, , or the upper bound of the continuous portion of the distribution is shrinking, since, (as shown in the proof of Theorem 3.2)
Furthermore, as goes to infinity, we see that goes to .
An illustration of the relationship between and the symmetric equilibrium values of and is given in Figure 1.
4 Search with Frictions
We turn our attention to the case where and/or . In the following result we establish sufficient conditions for there to be a symmetric equilibrium in pure strategies.
(Frictions) Suppose there are frictions, i.e. and/or . If then the unique symmetric equilibrium is one in which each box uses a fully revealing signal. For any fixed , there is some such that if the number of boxes is greater than or equal to , there exists a unique symmetric equilibrium in pure strategies, where each box chooses a fully informative signal.
We establish the result through the following two lemmata:
(Frictions) The reservation value assigned by to a box with a fully revealing signal is strictly higher than the reservation value assigned to a box with any other signal. Equivalently, if and then .
With a binary prior,
Consider any distribution , such that . Since , we must have , for some . Then,
Expression 4 is strictly increasing in and thus
Next, consider any distribution , such that , for some . Since , we must have . We have
Expression 4 is strictly increasing in , which is bounded above by and thus
where we used the fact that is strictly increasing in .
For a general prior, for any that is a mean-preserving contraction of ; which is intuitively clear666And proved as a lemma in Appendix C.1. since is more informative in the Blackwell sense than any . However, the fact that the reservation value assigned to the prior is strictly higher than for any other garbling relies on the binary prior assumption. Indeed, it does not hold in general for a prior with any other support.777A simple example of different fusions of a ternary prior that induce the maximal reservation value may be found in Appendix C.2.
(Frictions) If , then a symmetric vector of distributions is an equilibrium if and only if each distribution induces the maximal reservation value, .
Suppose there is a symmetric equilibrium in which each box chooses a distribution, that induces some . Suppose that box deviates and instead chooses distribution (, and ), such that . In other words, the mass of this distribution below its reservation value consists entirely of a point mass on . Since , we must have , for some . Then,
Expression 4 is strictly and smoothly increasing in and thus there is some such that for all , . Moreover, under this distribution it is clear that . Thus, box has achieved a profitable deviation. Note that for any it is easy to construct a corresponding .
If each box chooses then there is no profitable deviation, since any deviation will induce a strictly lower . That box will be visited last, and its payoff will be at most , by assumption. ∎
Combining the two lemmata yields Theorem 4.1. ∎
If the mean is sufficiently high, then it becomes paramount for a box to choose an experiment that entices Pandora into visiting it first. Since there are search costs, Pandora’s rule dictates that she visit more informative boxes first. Hence, in any proposed symmetric equilibrium other than the fully informative one, a box can deviate by choosing a distribution with a slightly higher reservation value, thereby ensuring that it is the first box visited. As a result of this incentive, there is an unraveling upwards to the distribution that induces the maximal reservation value, which is uniquely the prior distribution. Because the mean is so high, there can be no profitable deviation from the fully informative equilibrium. Crucially, any box that deviates from the fully informative equilibrium will guarantee that it is visited last. Then, because the mean–and equivalently the chance of a high realization under the fully informative signal–are so high, even though a deviating box can ensure that if it is visited it will be selected, the likelihood that Pandora ever visits that box is too low for the deviation to be profitable.
If, however, the mean, , is strictly below the cutoff () then this logic fails, since it is insufficiently likely that the box is of high quality. As in the high mean case, because of the incentive to be visited first, there cannot be a symmetric pure strategy equilibrium in which boxes are anything less than fully informative. However, because the mean is so low, there is no symmetric pure strategy equilibrium in which each box is fully informative. There, the likelihood that the deviating box will be visited is sufficiently high to ensure that the deviation is profitable. Indeed, we have
(Frictions) Suppose the prior distribution has support on and and let the mean satisfy . Then there is no symmetric equilibrium in pure strategies.
We sketch the proof (with formal detail left to Appendix C.3).
First, there can be no symmetric pure strategy equilibrium in which boxes play distributions with any . A box can always deviate by choosing a distribution that is almost identical but which induces a slightly higher . In doing so, the deviating box ensures that it will be inspected first, securing itself a first-order increase in its payoff (whereas any potential decrease in payoff resulting from the change in distribution will be second-order).
Second, because the mean is so low, there can be no equilibrium in which each box chooses the fully informative signal, since a box could deviate profitably to, say, a fully uninformative signal, and achieve a higher payoff. ∎
We can derive the following result regarding the symmetric mixed strategy equilibrium. We say that there is an induced mass point on a reservation value if there is a set of distributions that induce that is played with positive probability.
(Frictions) There is no symmetric mixed strategy equilibrium in which there is an induced mass point on any reservation value .
Recall that is the reservation value induced uniquely by a fully informative signal. The intuition behind this proof is similar to that for Theorem 4.4. Any box can deviate profitably by instead inducing a mass point on a slightly higher reservation value. In a sense this can be thought of an identical mixed strategy where the “ties are broken in box ’s favor”.
The detailed proof is included in Appendix C.4. ∎
This result arises as a consequence of the same sort of logic that dictates that sellers choose atomless distributions over prices in situations where their competitors can undercut, see e.g. [sta]. In our scenario, if there were a point mass on any reservation value other than the maximal reservation value, another box could “overcut” with information, and discretely increase its payoff.
Precise characterization of the mixed strategy equilibrium is a challenge. The utility of a box as a function of the vector of mixed strategies is quite complicated, due to the nature of Pandora’s stopping procedure. Even for the situation with just two boxes it is rather ungainly (see the proof of Lemma A.7, for the utilities in explicit form). Another difficulty lies in the set of pure strategies over which the boxes are mixing. That set–, the set of all probability distributions on with mean –is quite large, and the objective of solving each box’s respective maximization problem over this set has proved elusive. In a subsequent working paper building on this one, Au (2018) [au3] establishes some further properties that the mixed strategy must have. Moreover, he characterizes explicitly a mixed strategy equilibrium for the case where the searcher has a high outside option. This assumption allows for a simpler formulation of the boxes’ problem, enabling the result. For an outside option below this cutoff, that equilibrium; however, is not an generally an equilibrium.
5 Dynamics and Welfare Comparisons
We now derive the main result and contribution of the paper. Namely, we show how an increase in search costs can increase the welfare of the consumer.
5.1 Welfare Comparisons
Next, we compare the equilibria in the cases without and with search frictions.
5.1.1 The Case without Frictions
Let . If , and , then from Theorem 3.2, the unique symmetric equilibrium is for each box to choose distribution , defined as
where and ; and . Moreover, Pandora’s payoff from this search can be described as the random variable . With probability , takes value . Moreover, has a continuous portion supported on :
The expectation of is
Hence, the expected value to the searcher of , , is
5.1.2 The Case with Frictions
If there are frictions and , then Pandora’s expected payoff, , is
The main result:
Fix a binary prior with mean . Then, for any number of boxes such that , there exists a discount factor and a search cost such that if and , Pandora obtains a higher discounted expected utility from the case with frictions. If is finite, the frictionless case can lead to strictly higher utility for Pandora.
With frictions, the discounted expected utility of Pandora’s search is given by Expression 5.1.2. As and , Expression 5.1.2 converges to the expected value of the first order statistic of , i.e. the expected value of the maximum of draws from distribution :
This is strictly increasing in , and for finite , . Thus we must have . Finally, since Expression 5.1.2 is smooth and strictly increasing in and decreasing in , the result follows.
In Figure 2 we illustrate this dynamic for small and an expected reward of .888Note that in [down] the authors define a partial order as They then show (Theorem 2.2 [down]) that implies . However, here we do not use this result since we show a strict inequality.
We have shown that regardless of how many boxes there are, it is welfare improving to the searcher to be slightly impatient and/or have a small search cost. This slight cost leads to the “perfect competition” in information provision.
Because Pandora’s search is directed, it becomes important for the boxes to choose a signal that entices Pandora to visit them first. Crucially, this choice occurs before Pandora commences her search, and this, in tandem with the search cost, forces the boxes to compete on information.
In this paper we look at the problem of information provision in a sequential, directed search setting. With frictions, the choice of whom to visit first becomes paramount to the searcher. Each box, all else equal, would prefer to be visited first, and this affects its choice of signal. With frictions, as long as the expected value of the prize is sufficiently high, the unique equilibrium is one in which each box chooses a signal that is fully informative. This can be contrasted with the unique equilibrium in the case without frictions, which, for finite , is never fully informative. The searcher, Pandora, naturally prefers more informative signals, and so as long as the costs are sufficiently low, she counter-intuitively prefers the case with frictions (see Figure 2).
One alternative insight or interpretation of the results in this paper is that consumers gain when sellers are forced to compete on order. Another way this effect could be derived is by considering equilibria that are not fair. If in the frictionless case Pandora is allowed to choose to visit more informative boxes first (violating the fairness assumption), then competition on order will arise, and boxes will be completely informative. However, as we show in this paper, search frictions provide this impetus, even when Pandora’s search policy must be completely fair.
Appendix A Appendix (Section 2 Proofs)
a.1 Theorem 2.8 Proof
In this proof, we show equilibrium existence for a general prior, and not just one supported on two points. Because of that, the set of strategies is , the set of all mean preserving contractions of the prior. If the prior is binary, . We first prove the following proposition:
For each box , the set of pure strategies, , and the set of mixed strategies, , are compact, convex and Hausdorff.
Our proof proceeds through a series of Lemmata. First, from Elton and Hill (1993) [hill], we have
The set of fusions of , , is closed with respect to the weak-* topology.
(Theorem 3.11 in [hill]) is convex.
Using these results, we have
The pure strategy set for a box , , is compact (with respect to the weak-* topology) and convex. Moreover, the Cartesian product of the pure strategy sets, , is compact and convex.
From Lemma A.3, is convex. Denote by the set of all Borel probability measures on a compact set . If is a probability measure on , then . Using the Riesz Representation theorem one can show that is weak-* compact. By construction, is closed and since it is a subset of it must therefore be compact.
Convexity of the Cartesian product of the pure strategy sets, , is immediate. Moreover, by Tychonoff’s theorem, is compact. ∎
Since the weak-* topology is a Hausdorff space, we have
The pure strategy set for a box , , is Hausdorff.
The set of mixed strategies, , is compact, convex and Hausdorff.
By construction, is convex. Denote by the product set of box ’s pure strategies. is endowed with the product topology and the product sigma-algebra. By Tychonoff’s theorem, we may immediately conclude that is compact (with respect to the product topology). Then, let be the set of all Borel probability distributions on . Again, by the Riesz Representation theorem, this set is weak-* compact. ∎
Naturally, is also compact, convex, and Hausdorff. ∎
Armed with these results, to prove Theorem 2.8, we use results developed in Reny (1999) [reny].
By assumption, is bounded and measurable. From Lemma A.5, the set of strategies is Hausdorff. In the terminology of [reny], this game is a compact, Hausdorff game. The set of Borel probability measures on is itself compact in the weak-* topology. Moreover, from Proposition 5.1 in [reny], the mixed extension of the game , , is reciprocally upper semicontinuous. Next, we show the following Lemma
For each of a box’s pure strategies, its payoff is lower-semicontinuous in its opponents’ mixed strategies.
For expository purposes, we first illustrate the proof for the two box game, which is much simpler; likewise through symmetry, we may focus on box without loss. Let box choose and box choose . It is clear that box ’s payoff is not lower-semicontinuous in box ’s pure strategies. We wish to show that for each of box ’s pure strategies, its payoff is lower-semicontinuous in the mixed strategies of box .
First, note that the set of pure strategies is set , which is a subset of the set of probability measures on the metric space with mean . The set of pure strategies is equipped with the weak- topology, and correspondingly is endowed with the Lévy-Prokhorov metric, which is defined as follows. For any subset define the -neighborhood, , of as
Then, for any Borel probability measures , the Lévy-Prokhorov metric, is defined as
The set of mixed strategies is, in turn, the set of probability measures on the metric space , which is also equipped with the weak- topology and endowed with the Lévy-Prokhorov metric. For the sake of completeness, we write this out. For any subset , define the -neighborhood, , of as
Then, for any Borel probability measures , the Lévy-Prokhorov metric, is defined as
Next, we write out box ’s payoff function explicitly: Suppose box chooses distribution and box distribution . Then, box ’s payoff is:
Let box choose mixed strategy over distributions (which has a point mass of weight on ). Then, box ’s payoff from a pure strategy is
where is the set of distributions in the support of that induce a and similarly for and . is given analogously. We take the difference and is
Clearly, , , and are each less than . Hence, if then Expression A.1 is itself less than . The result (for two boxes) is shown. Furthermore, it is easy to see how this result generalizes to boxes.
Write and as the vectors of mixed strategies for the boxes other than box . Under this vector of mixed strategies, there are a finite number of permutations, , of the realized orderings of values. Index each permutation by a unique , , and define in the same manner as in the two box case. Additionally, for each , write box ’s payoff as . Thus, is
which is less than if . ∎
It is immediately clear that lower-semicontinuity in the opponents’ mixed strategies for each of a box’s pure strategies implies payoff security on the set of mixed strategies. Then, since is payoff secure and reciprocally usc, by Corollary 5.2 in [reny] it is better-reply secure, and therefore, also by Corollary 5.2, must have a mixed strategy Nash Equilibrium. ∎
Appendix B Appendix (Section 3 Proofs)
b.1 Proposition 3.1
First, we establish the following lemma:
There are no symmetric Nash Equilibria where boxes choose discrete distributions supported on points.
It is easy to see that there is no symmetric equilibrium in which each box chooses a distribution consisting of a single point mass. Such a distribution could only consist of distribution with weight placed on and would yield to each box a payoff of . However, there is a profitable deviation for a box to instead place weight on and weight on (). In doing so, this box could achieve a payoff arbitrarily close to .
Now, assume . Observe that a strategy consists of a choice of probabilities , , and support such that .
The expected payoff to each box from playing an arbitrary strategy, , is
We claim deviating to the following strategy is profitable: where is played with probability and is played with probability , for , where (Again, ).999Note that we can always find such an . The expected payoff to box 1 playing strategy is
Note that the deviation is profitable for box 1 if
Which holds for a sufficiently small vector .
Then, we extend this lemma to show that there can be no distributions with point masses on any point in . Using an analogous argument to that used in Lemma B.1, it is easy to see that there cannot be multiple point masses. Accordingly, it remains to show that there cannot be a single point mass. First, we will show that there cannot be an atom at any point . Suppose for the sake of contradiction that that there is a symmetric equilibrium where each box plays a point mass of size on point . That is, each box plays strategy that consists of a distribution and a point mass of size on point . Let . Then, box ’s payoff is
Then, let box deviate by introducing a tiny point mass of size at and moving the other point mass to and reducing its size slightly to (); call this strategy . The payoff to box is
Suppose that this is not a profitable deviation. This holds if and only if
Clearly, as and go to zero we achieve a contradiction. Hence, there is a profitable deviation and so this is not an equilibrium. It is clear that there cannot be an equilibrium with a point mass on and so we omit a proof. ∎
b.2 Theorem 3.2 Proof
We prove this theorem for the case where . The remaining case, , is proved analogously and the doubtful/incredulous reader is directed to the working paper [Hulko], which contains the detailed proof.
b.2.1 Equilibrium Existence
First, we show that this is an equilibrium. Accordingly, we need to show that there can be no unilateral profitable deviation. Define as , which, recall has a point mass on . Moreover, define as the corresponding continuous portion of the distribution of ; :
with associated density
Evidently, it suffices to show that our candidate strategy achieves a payoff of at least to the box who uses it, irrespective of the strategy choice by the other boxes. Suppose for the sake of contradiction that there is a profitable deviation, that is, box deviates profitably by playing strategy . Clearly, we can represent as having a point mass of size , on (naturally, if , then there is no point mass there). Written out, consists of
and . Define . Naturally, . Then, box ’s utility from this deviation, , is101010Note that the first term, , is derived below, in the proof of Lemma 3.3.
Evidently, this is a profitable deviation if and only if ; that is,
where we used the fact that and . It is clear that and thus we have
We have established a contradiction and thus the result is shown.
b.2.2 Equilibrium Uniqueness
First, we tackle Lemma 3.3:
Lemma 3.3 Proof
Let each box play strategy where they each put weight on . Suppose that box deviates and plays strategy consisting of random variable distributed with value with probability and with probability .
Then, box ’s payoff is
This must be less than or equal to :