Consumer search costs and market performance.

Davis, Douglas D. ; Holt, Charles A.

I. INTRODUCTION

One of the seminal developments leading to the information economics revolution in the 1970s and 19c0s was Diamond's [1971] theoretical result that, in the absence of publicly posted price information, the existence of even a small search cost could lead to monopoly pricing. The intuition is straightforward: No buyer with one price quote would want to search for a second, unless it is likely that the buyer would encounter a price reduction that covers the search cost. Thus each seller has an incentive to price slightly above any common price, and the noncooperative equilibrium in a single-stage game yields a monopoly price. This result is viewed as a paradox, since a "small" search cost produces high prices, but a zero search cost would produce the usual Bertrand incentives that drive prices to competitive levels, in the absence of capacity constraints and other imperfections.(1)

Recent theoretical work has focused on finding a "resolution" to the paradox. Stahl [1989], for example, generates a smooth transition between competitive and monopoly outcomes as the fraction of consumers with zero search costs increases from zero to one. Interestingly, Stahl finds that increasing the number of sellers makes pricing more monopolistic, holding the number of informed buyers constant. Bagwell and Ramey [1992] propose an alternative resolution that applies to an infinitely repeated market game where sellers who raise prices can develop a reputation that can affect sales in future periods. Bagwell and Ramey show that buyers can obtain lower equilibrium prices by following a "loyalty-boycott" search rule, where low-pricing sellers are rewarded with repeated purchases, and high-pricing sellers are punished with switching. The price predictions vary continuously between monopoly and competitive levels, depending on the size of the consumer search cost.

The policy relevance of the Diamond paradox depends on whether there are realistic market environments in which small reductions in the availability of price information can cause large price increases.(2) Due to the difficulty of controlling and measuring information flows in natural markets, the laboratory represents an ideal place to evaluate the Diamond paradox and proposed resolutions. The only experimental analysis of these issues is reported by Grether, Schwartz and Wilde [1988], who observed monopoly pricing in three of four predicted cases. These results are suggestive, but not definitive, for reasons discussed below. In particular, as Grether, Schwartz and Wilde [1988, 328] observe, all four cases involved the same group of subjects "to the extent possible."

This paper reports an experiment consisting of twelve market sessions designed to assess the effects of public price information. The markets are conducted as normal posted-offer markets, except that buyers must pay a small cost each time they approach a different seller. In baseline "posted-offer" treatments, all prices are publicly displayed to buyers, while in "search" treatments, prices are not publicly displayed. In brief, we find that prices approach competitive levels under the posted-offer treatment, and prices are significantly higher under the search treatment. Nevertheless, the Diamond prediction of monopoly pricing under search is not observed, although increases in search costs do raise prices. These results indicate that economists should take a more careful look at attempts to resolve the Diamond paradox by introducing other factors that may impede monopoly pricing in markets with consumer search.

The paper is organized as follows. Our two primary treatments are outlined in section II, and results are presented in section III. The failure to observe the Diamond monopoly price prediction motivated additional research on the effects of reputations and search costs, which is presented in section IV. The final section contains a conclusion.

II. EXPERIMENT DESIGN AND PROCEDURES

The first six sessions to be discussed involve the posted-offer and search treatments mentioned above. Each session consisted of two sequences of twenty market periods, with prices being posted publicly to buyers in one sequence and not in the other. Sellers could not see one another's posted prices in either treatment. The treatment order was alternated in every other session in this "within-group" design. In all sessions the buyers' shopping cost was set at fifteen cents per seller approach.

Sessions were conducted as standard posted-offer markets, with the exceptions of the shopping cost and the privacy of price information in the search treatment. At the outset of each trading period, buyers and sellers are provided unit values and costs. Seller earnings accrue as the difference between the price and the cost of units sold. Buyer earnings are the difference between the value and the price of units purchased, less all seller-approach expenses. At the beginning of each trading period, sellers privately make price (and maximum sales quantity) decisions, while buyers wait. Once all sellers have completed their decisions, buyers are randomly drawn one at a time and given the opportunity to make as many purchases as they want at the posted prices. The period ends when all buyers have been given an opportunity to shop, or when all sellers are out of units. Consistent with the bulk of market experiments, all cost and value information was private and was revealed to none of the other traders.(3)

The search treatment implements a standard sequential search setup: in order to see a seller's price, the buyer must specify the seller and pay the fifteen-cent cost. The buyer may then either purchase at the posted price by pressing "p" on the keyboard, or shop elsewhere by pressing "s." In the posted-offer treatment, a buyer could see all of the sellers' prices at no cost, and the displays indicated whether a seller was out of stock. The buyer would still have to pay the fifteen-cent shopping cost to approach a seller. This cost is like a travel cost in the sense that it is independent of the number of units purchased. Although the shopping cost is incurred in each treatment, it becomes a search cost when prices are not public in the search treatment.(4)

Each session consisted of three buyers (B1-B3) and three sellers (S1-S3), with values and costs indicated by the trader identifications for each unit on the market demand and supply curves shown in Figure 1. The buyers are identical, each with a marginal value of one unit at ninety-five cents, a second unit at fifty-five cents, and then zero for all additional units. Similarly, sellers faced identical costs, with a single unit at a marginal cost of zero, and three additional units with costs at twenty-five cents.(5) The competitive prediction is [P.sub.c] = 25 cents.

Although the competitive equilibrium is a useful benchmark, it is not a Nash equilibrium for the market-period "stage-game" in this design. To see this, observe that half of the units are not sold at a common competitive price of twenty-five cents. Thus, sellers do not earn the twenty-five cent profit on the low-cost unit with certainty, and any seller may increase expected earnings by offering a single unit for a price slightly less than twenty-five cents. Also, there is no Nash equilibrium at prices below twenty-five: For any set of prices below twenty-five cents, only three units are offered, and any seller could unilaterally increase earnings by selling four units at a price of fifty-five cents.

The Nash equilibrium involves mixing over the range from twenty-five to fifty-five. The observation that competitive and Nash predictions differ here is useful, since it is well documented that in posted-offer type markets prices tend to be drawn away from the competitive outcome when the predictions differ.(6) Nevertheless, as a behavioral matter, the absence of a pure strategy Nash equilibrium at [P.sub.c] should be of little consequence, since the bulk of the pricing density in the mixed equilibrium distribution clusters about twenty-five. The median of the pricing distribution, for example, is 25.73 cents (see Appendix A). This median is far below the joint-profit-maximizing ("monopoly") price of [P.sub.m] = 55 cents.(7)

This market design does not exactly implement any specific search model in the literature. In particular, the design violates the standard assumption that sellers produce at constant marginal cost and that buyers have a constant and uniform reservation value. The high-value and low-cost steps were added to give the competitive and monopoly predictions a realistic chance of being observed, since stable outcomes in which one side of the market earns nothing are rarely observed. The design is anchored on a minimum earnings of twenty-five cents per trader per period at each of these outcomes. For this reason, the cost step for the first unit is twenty-five cents below [P.sub.c] = 25. Similarly, the placement of each buyer's first unit value at forty cents above the monopoly price, [P.sub.m] = 55, guarantees each buyer a minimum earning of twenty-five cents at the monopoly outcome, after subtracting out the fifteen-cent shopping cost.

Nevertheless, our design retains the features necessary to generate starkly different equilibrium predictions across price-information conditions. In contrast to the prediction that publicly observed prices will cluster about the competitive prediction, the Nash equilibrium for the static search game is for all sellers to offer their units at [P.sub.m] = 55. To verify that this is a Nash equilibrium, assume that buyers randomly approach the sellers in light of a common expectation that all sellers will post a price of fifty-five cents. Since sellers are approached randomly, each may sell between zero and four units. It is straightforward, if somewhat tedious, to show that under these conditions expected profits are about 78.5 cents. (See Appendix B for details.) To demonstrate that this is a Nash equilibrium, first note that price deviations are not observed ex ante by buyers in the search treatment. Therefore, no unilateral reduction from fifty-five cents can be profitable, since the price decrease will neither divert purchases from other sellers nor allow for an increase in aggregate sales. Unilateral price increases are also unprofitable: Given that buyers expect sellers to post a price of fifty-five cents, any deviation more than fifteen cents above fifty-five will induce a buyer to shop elsewhere. Thus, the maximum possible deviation is a price of seventy cents, in which case expected earnings are about 65.2 cents (see Appendix B). As shown in Appendix C, this equilibrium is unique in the set of equilibria where all sellers have positive expected sales.(8)

Two additional features of our design bear comment. First (and unlike Diamond's original formulation of the problem), our implementation is based on an explicitly static analysis. Any failure to observe monopoly prices may be attributable to omitted dynamic considerations, such as learning, or the possibility that buyers are able to keep prices low by exploiting the repeated nature of the market game to punish high-pricing sellers. Second, we wish to emphasize the differences between our implementation and that reported by Grether, Schwartz, and Wilde [1988]. In their search ("monopoly") treatment, buyers were shown the complete list of prices actually posted in each period, without seller identifications. A buyer could avoid a search cost by making a purchase (if profitable) from one seller randomly selected by the experimenter. Alternatively, the buyer could pay a cost (ten cents) to obtain a sample of two or more randomly selected prices, so that a purchase could be made at the lower price. We decided not to reveal any prices to buyers unless they paid a search cost in a sequential search setup. We were motivated in part by Stahl's [1989, 7001 argument against "the dubious assumption that consumers can 'see' deviations by firms before they actually search."(9)

Subjects were University of Virginia students who were recruited from economics classes. Buyer and seller roles were determined by random draws. Then subjects were seated at visually isolated personal computers. Instructions were presented on the displays as an experimenter read them aloud.(10) After the initial twenty period treatment, supplemental instructions were read for the final twenty-period sequence. The final period was not announced for either sequence.(11) Subjects were paid $6.00 for showing up, in addition to earnings from trading. Earnings averaged $24.10 per subject and ranged from $12.50 to $35.25. Payments were made in private immediately after the session. Two of the six sessions were conducted with subjects who had previously participated in a laboratory posted-offer market session (but of a different design), while participants in the remaining four sessions had no previous experience in an economics experiment.

The configuration of treatments by session is summarized by the three-part identifiers in the first column of Table I. Each identifier consists of a two-letter prefix ("SP" or "PS") to indicate whether the search or posted-offer sequence came first, followed by a number indicating the order in sequence, and an "x" if the session used experienced participants. Thus, for example session PS3x in Table I refers to the third session in the posted-offer/search sequence, with experienced participants.

TABLE I

Mean Deviations of Transactions Prices from the Competitive Prediction for the Final 5 Periods: Search and Posted-Offer Treatments

 Search Sequence Posted-Offer Sequen

Session P-[P.sub.e] P-[P.sub.e]
 SP1 5 1
 SP2 22 1
 SP3x 25 1
AVG. for SP1-3 18 1
 PSI 15 6
 PS2 3 1
 PS3x 14 8
AVG. for PS1-3 11 5

III. RESULTS

Mean transactions price paths are shown on the right side of Figure 1. Results for the search/posted-offer treatment sequence are shown above the posted-offer/search results.(12) The bold line in each panel represents the mean price path averaged across the three sessions for each treatment sequence. Mean transactions prices for the individual sessions are indicated by the light lines. In each panel, the thin solid line represents a session with experienced subjects. Several conclusions are immediately apparent from Figure 1. First, prices exhibit a much more pronounced tendency to converge to the competitive level in the posted-offer treatment than in the search treatment. In fact, prices are higher in the search sequence of each session than in the corresponding posted-offer sequence. Table I shows mean deviations from the competitive price for the last five periods each session.(13) Mean price deviations in the search sequence of each session (second column) are larger than the mean price deviations for the corresponding posted-offer sequence (third column). This outcome allows rejection of the null hypothesis of no treatment (public price-posting) effect at a 98 percent confidence level, using the nonparametric randomization test.(14) This observation supports our first conclusions:

CONCLUSION 1. Search matters: nonpublic postings raise prices.

This conclusion is not particularly surprising, and was previously observed by Grether, Schwartz and Wilde [1988]. The relationship of transactions prices to the monopoly prediction, [P.sub.m] in the search treatment is perhaps more interesting. When search precedes posted offer (the left side of the upper panel), the bold overall price path is roughly two-thirds of the distance from the competitive to the monopoly price. In contrast, overall average transactions prices are only about one-third of that distance in the search treatment when it follows posted offer (the right side of the lower panel). Moreover, even "near monopoly" prices are the exception rather than the rule. In more than half of the six sessions summarized in column 2 of Table I, mean prices are as least as close to the competitive level as to the monopoly outcome. This forms our second primary conclusion:

CONCLUSION 2. The Diamond-paradox monopoly price is not consistently observed in this environment.

The variability of performance within treatments suggests that differences in participant behavior may be important.(15) Even in a single session there is often a lot of price dispersion, which may impede an upward price-adjustment process. It may be straightforward for sellers to appreciate the logic of raising price above some common level, as long as the price increase is less than the shopping cost. But when the prices of other sellers are unobserved (and are in fact very dispersed), sellers may find it very risky to raise price, since the sellers know little about buyers' price expectations. Buyers, on the other hand, may be able to force prices down by "punishing" sellers. As suggested by Bagwell and Ramey, buyers may either refuse to purchase if a posted price is unacceptably high, or may purchase but shop elsewhere in other periods.

There is at least anecdotal evidence that buyers employed punishment behavior of this sort. Consider for example, the sequence of contracts for the search sequence of session PS3x in Figure 2. Data for the twenty trading periods are separated by vertical lines. Within periods, price postings for sellers S1, S2 and S3 are represented, in respective order, by crosses (+). Contracts for single units are denoted by small dots ([multiplied by]); multiple units sold at the same price show as an overlap of dots to the right of the price postings. Thus, for example, period 1 of session PS3x is illustrated between the left-most pair of vertical bars in Figure 2. In this period, sellers S1 and S2 posted prices five cents below [P.sub.m], and S3 posted a price twenty cents above [P.sub.m]. Sellers S1 and S2 subsequently sold two units each, while S3 sold a single unit.

Seller S2 continued to post prices well above [P.sub.m] in periods 2 to 5 and posted prices at [P.sub.m] in periods 7 and 10. Although this seller occasionally made sales at high prices (e.g., in periods 2, 3 and 7), the buyers became wary of these high postings and failed to approach this seller in periods 10 through 17, this despite successive price decreases in periods 11, 14 and 17 (where price was lowered to the cost of the low-cost unit). Although punishment was not immediate in this session, seller S2 clearly paid for high price postings in early periods.

Instances of punishment were also observed in the other sessions, albeit less dramatic. It is difficult to develop a reasonable statistic for switching away in response to high prices, since, given shopping costs, a high price in a given period is a subjective assessment on the part of the buyer. In Figure 2, for example, one buyer (B1) consistently purchased units from S1, despite the fact that this seller posted the highest price in periods 11 to 20.

One crude means of assessing the tendency for buyers to punish sellers posting high prices is to calculate the propensity of buyers to switch away from sellers after observing prices in various ranges. Figure 3 summarizes this information for the search sequences of our initial six sessions. In the figure, each bar reflects the percentage of the time that buyers in a particular session observed prices in the indicated price range and then switched. Thus, each cluster of six bars summarizes behavior over a given price range, for all six sessions. The solid line connects dots representing average behavior in each cluster. From the within-period behavior summarized in the left panel of the figure, it is seen that buyers tend to pay the fifteen-cent approach fee more than once in a period only if prices are very high, particularly above the fifty-five-cent monopoly level. But, as indicated by the right panel of the figure, except at the very highest prices, buyers exhibit much more willingness to shift among sellers between periods. Behavior is highly variable across sessions. Moreover, the high propensity of buyers to switch sellers across periods even in the lowest price range suggests that some of the switching is due to price-searching rather than intended punishments. Nevertheless, even though the effect is clearly not dominant, the generally positive slope of the lines summarizing average switching propensities in each panel suggests that reputations may play some role in determining pricing, since buyers exhibit at least some tendency to go elsewhere as prices increase.

The effects of reputation may be more systematically evaluated by examining behavior in additional laboratory sessions where reputations are controlled. A primary goal of the research described in the next section is to isolate the effects of sellers' reputations.

IV. SEARCH COSTS AND REPUTATIONS

Although the absence of price information tends to raise prices in an environment where shopping is costly, prices are not raised to monopoly levels as implied by the Diamond paradox. This raises the question of whether there is a reasonable baseline condition in which the absence of public price information clearly generates monopoly prices. To evaluate this question we tried two new treatments, with three sessions each. In both cases, we attempted to control for reputations by disguising seller identities. The second treatment differs from the first in that shopping costs are doubled. These sessions are summarized by the identifiers listed in the first and third columns of Table II. These identifiers roughly follow the labeling convention in Table I: the two-letter "SN" prefix indicates that price information was not publicly displayed (e.g., "search"), and that sellers' reputations were disguised ("no reputations"). This prefix is followed by an "L" (search cost = 15 cents) or "H" (search cost = 30 cents), a number in sequence, and an "x" if experienced participants were used.

TABLE II

Mean Deviations of Transactions Prices from the Competitive Prediction for the Final 5 Periods: Search with No Reputations

Search, No Reputations, Search, No Reputations,
c = 15 c = 30
Session P-[P.sub.e] Session P-[P.sub.e]
SNL1 18 SNH1x 26
SNL2 6 SNH2x 31
SNL3x 15 SNH3x 21
AVG. 13 AVG. 26

Controlling for Reputations

The most important difference between the search treatment described in the previous section and the SNL treatment listed in the first and second columns of Table II is that seller identities were disguised so that buyers would be unable to determine which sellers posted which prices. Higher prices in this treatment than in the preceding treatments would indicate that reputation effects in fact tend to lower prices.(16)

To disguise seller identities, the following procedures were used. At the beginning of the session, sellers were visually isolated from the buyers and were given a colored marble "identifier." Prior to each period a monitor drew the marbles in sequence from the urn and assigned the role of seller S1 to the first marble drawn, seller S2 to the second marble, and S3 to the third marble. The sellers then took their seats and posted prices. Further, to prevent sellers from divulging their identity via very rapid or very slow price postings, the monitor made the final price confirmation for each seller, once all sellers had finished posting prices. In this way, the posting sequence was terminated in a preannounced, anonymity-preserving fashion.

In an effort to give monopoly price outcomes a reasonable chance, we made two additional procedural changes. First, given the likelihood of sequencing effects, we did not precede any of the sessions with a posted-offer sequence.(17) Second, we decided to let sellers see each other's price postings. While the relevant theory is silent on the matter of the amount of price information available to sellers, we decided to give this information to sellers in an effort to facilitate learning of the recursive "price-plus-shopping-cost" reasoning that goes into generating a static equilibrium at [P.sub.m] = 55 in the search treatment. If sellers can see the prices posted by the others, then the sellers at the low end of the distribution may feel more secure about price increases that are small relative to the shopping cost. In all other respects procedures were identical to our initial search treatments: shopping costs were fifteen cents per approach, each sequence consisted of twenty periods, and the experience profile consisted of two sessions with inexperienced participants and a single experienced session.

As in the preceding section, results are obvious and follow almost without comment from the mean transactions price data in the second column of Table II. Also, it is apparent from the left panel of Figure 4 that prices do not approach monopoly levels when reputations are disguised. In fact, comparing price data in sessions SNL1-3x (second column of Table II) and SP1-3x (second column of Table I), one can see that mean prices were actually five cents lower on average in the no-reputation sessions than in the reputation sessions. Although results are too mixed to allow us to make a reasonable statistical claim that prices are actually higher when seller identities are not hidden, the results provide absolutely no support for the hypothesis that disguising information tends to raise prices. This motivates our third observation:

CONCLUSION 3. Seller reputations do not lower prices in this context; prices do not approach the monopoly prediction, even when seller identities are disguised and sellers can see each others' prices ex post.

The failure to observe monopoly price outcomes, even when seller identities are hidden and when sellers can see the prices posted by others, suggests that sellers fail to understand the recursive "price-plus-search-cost" reasoning underlying our static implementation of the Diamond result. In large part this may be due to initial price dispersion, which makes price increases appear risky. Nevertheless, search costs provide sellers with some pricing discretion, much in the manner of switching costs or other entry impediments. To the extent this is true, the magnitude of the impediment determines the extent of sellers' price discretion. As a final treatment, we examine this prediction by increasing search costs.

Search Cost Increases

Our final treatment consists of three sessions conducted as in the no-reputation treatment, except that search costs were increased to thirty cents. With minor exceptions, other procedures were exactly as described in the reputation control sessions.(18)

Results of these "SNH" sessions are summarized in the right panel of Figure 4 and in the fourth column of Table II. A comparison of the left and right panels of Figure 4 indicates that increases in search costs raise transactions prices. In the high-search-cost sessions, the mean deviation of transactions prices from [P.sub.e] is twenty-six cents, twice the thirteen-cent deviation observed in the low-search-cost sessions. Further, the mean transactions price in each SNH session is higher than the highest price in the SNL treatment. The uniformity of results allows rejection of the null hypothesis that increasing search costs does not affect prices at a 95 percent confidence level using the nonparametric randomization test.(19) This leads to our fourth, and final conclusion.

CONCLUSION 4. Increases in search costs tend to increase prices, at least when seller identities are disguised.

V. DISCUSSION

This paper has examined the results of twelve market sessions designed to evaluate the behavioral robustness of the Diamond prediction. The two primary lessons of this research are (1) that the absence of public price information raises prices when shopping is costly, but (2) that the monopoly prices implied by the "Diamond paradox" are not generally observed. Efforts to find a baseline treatment with monopoly price outcomes yielded two additional conclusions: (3) controlling for seller identities is not sufficient to generate monopoly prices, and (4) increases in the search costs will raise prices.

In an important sense, the results of our initial sessions are consistent with the intuition motivating Bagwell and Ramey's efforts to resolve the Diamond paradox. The implied thrust of their work is that the "Diamond Paradox" is too extreme, and that in richer contexts non-monopoly prices will be observed. But equally important, results of our latter sessions are also inconsistent with the reasoning underlying Bagwell and Ramey's theoretical modifications. We observe non-monopoly prices, even when reputations cannot form.

What then causes below-monopoly prices? We conjecture that, as a behavioral matter, the Diamond result breaks down because sellers neither immediately appreciate nor do they readily learn the recursive "price-plus-shopping-cost" reasoning necessary to generate the monopoly outcome. In the standard posted-offer environment sellers may simply face too much price dispersion to know when a price increase is not likely to cause buyers to shop elsewhere. But search costs do give the seller some market power, in that they make it difficult for buyers to leave a seller. Thus, the higher the search cost, the more market power sellers possess.

Our future research in this area will proceed in two directions. The first involves examining just where and how the Diamond prediction breaks down. It is possible that there are some contexts in which sellers are likely to raise price to monopoly levels in a series of small increments. One possible approach is to evaluate the differences between our treatments and the Grether, Schwartz and Wilde design (where monopoly prices were observed). The most obvious alterations are adjustments that facilitate the learning process. For example, we might consider a treatment where sellers are informed of the monopoly price, and where both buyers and sellers know the distribution of prices each period. Other possible alterations include increasing the number of buyers, decreasing the capacity of buyers, and perhaps prohibiting seller stock-outs in a period.(20) The second direction for future research involves evaluation of predictions in the switching cost literature. Costs incurred when a buyer changes sellers may generate price outcomes consistent with those observed here, even when postings are public.(21)

APPENDIX A

Equilibrium Mixing Distributions for the Posted-Offer Treatment

The mixed-strategy equilibrium is calculated by identifying the pricing distributions that any two of the sellers must follow in order to make the third seller indifferent between all prices in the range of randomization. These calculations will assume risk neutrality and a continuous price distribution. It is apparent from Figure 1 that the only profitable unit at a price of twenty-five cents is the seller's zero-cost unit. The equilibrium strategies to be constructed will generate prices on the interval [25, 55], with only one unit offered at twenty-five cents and with all four units offered at higher prices. (The restriction to one unit offered at a price of twenty-five cents is made to avoid a continued reference to sellers pricing slightly below twenty-five cents.) A unit priced at twenty-five cents will, therefore, always sell since at most three units will be offered at that price, and demand is six in this range. The earnings of a (zero-cost) unit priced at twenty-five will be twenty-five cents, and this provides the "security earnings" level. The expected profit for any other price in the mixing range must also be twenty-five cents. Defining F(p) as the probability that a seller prices below p, and F(25) as the probability mass point at the competitive price, any seller will mix if other sellers price so that

(A1) 25 = [F(25).sup.2](4p - 75) + 2F(25)[F(p) - F(25)]p

+ 2F(25)[1- F(p)](4p - 75)

+ 2[1- F(p)][F(p) - F(25)](2p - 25)

+ [[1- F(p)].sup.2](4p - 75).

Each of the additive terms in equation (A1) is the probability of a particular outcome, weighted by the outcome's payoff. The leftmost term on the right side, for example, is the probability that both of the other sellers price at twenty-five cents in order to sell their single, low-cost units with certainty. In this case, a seller posting a higher price p will sell four units, at a sales cost of seventy-five cents (twenty-five cents for each of the three high-cost units). Similarly, the second term on the right corresponds to the two possible ways that one of the other sellers sells a unit at twenty five cents, and the other seller sells four units at price between twenty-five and p, which leaves one unit demanded at the price p.

To calculate the mass point at twenty-five cents, F(25) note that F(p) = 1 at the limit price p = 55. Thus, (A1) evaluated at p = 55 yields

25 = [F(25).sup.2]145 + 2F(25)[1-F(25)]55.

Solving, F(25) = .212857.

F(p) is most easily solved implicitly. Collecting terms by payoffs, (A.1) may be rewritten as

(A2) 25 = A(4p - 75) + Bp + C(2p - 25),

where A = [F(25).sup.2] + 2F(25)[1 - F(p)] + [[1 F(p)].sup.2]; B = 2F(25)[F(p) - F(25)]; and C = 2[1 F(p)][F(p) - F(25)]. Since p affects A, B, and C only through F(p), equation (A2) can be solved for p as a function of F(p).

(A3) p = (25 + 25C + 75A) / (4A + B + 2C)

which is the inverse of the probability distribution function. From equation (A3) it can be seen that in this mixing equilibrium most prices will be very close to the competitive price. For example, to find the median, set F(p) = .5 (and F(25) = .216609) in A, B and C, in equation (A3), which yields p = 25.73 cents, just above the twenty-five-cent competitive price.

APPENDIX B

The Existence of a Nash Equilibrium at [P.sub.m] = 55 under the Search Treatment

In the candidate equilibrium, all sellers offer four units at fifty-five. Thus, twelve units are offered in aggregate, and six are demanded. Given the excess supply, it is necessary to calculate earnings in expectation. First, consider expected earnings at a common price of fifty-five cents. For specificity, consider sales for seller S1. At fifty-five cents, each buyer is willing to purchase two units from any seller, and each seller has a maximum capacity of four units. Thus, S1 may sell four units (and earn 145 = 4 X 55 - 3 X 25); two units (and earn 85 = 2 X 55 - 25); or zero units (and earn zero). Without loss of generality, assume B1 shops first, B2 second and B3 third. The possible selection sequences for buyers B1 and B2 are summarized in the upper part of Figure 5: buyer B1 may approach S1, S2 or S3, and for each selection by B1, buyer B2 may also approach S1, S2 or S3. Choices for buyer B3 in the case that all sellers post the same price are listed in the middle part of the figure. As indicated in the figure, B3's choices are similar to those for B1 and B2, except in the event that B1 and B2 approach the same seller, in which case that seller is out of stock and unavailable.

Quantity outcomes for S1, and the associated probabilities, are summarized below B3's choices, in the middle of the figure. For example, the 1/18 probability listed on the left side of the row is the probability that B1 chooses S1, times the probability that B2 chooses S1 (thus exhausting S1's supply), times the probability that B3 chooses S2 instead of S3: 1/3 X 1/3 X 1/2 = 1/18. The "4" listed below the "1/18" in the figure represents the sales quantity for S1, in this case four units. Seller S1's profit for this case (not shown) is 155: 220 from selling four units at fifty-five cents less seventy-five for three high-cost units at twenty-five cents each. Weighting S1's profit for each outcome by its probability, it is straightforward but tedious to show that expected earnings for S1 are 78.519 cents.

Now consider the profitably of deviations from fifty-five cents. As argued in the text, no unilateral price reduction can increase expected earnings: since all sellers offer four units, at any price of fifty-five cents or less a reduction will only reduce revenue per unit, leaving the probability of sales constant. Thus expected earnings must fall with any unilateral price decrease.

Expected earnings also fall with any unilateral price increase. Although price increases raise revenues per unit, they also affect the sales probabilities in a way that reduces expected earnings, since at any price over fifty-five cents a buyer will purchase only one rather than two units from a seller. This latter effect is attenuated somewhat, however, since it is possible to sell units to all three buyers. The net effect depends on the size of the deviation.

Consider the maximum possible positive deviation, a fifteen-cent increase to a price of seventy cents. (Recall that any deviation larger than the search cost will motivate buyers to search elsewhere.) Again for specificity suppose seller S1 deviates, with S2 and S3 each posting fifty-five cents. At a price of seventy cents each buyer approaching S1 will purchase a single unit. Thus, S1 may sell three units (and earn 160 = 3 X 70 - 2 X 25); two units (and earn 115 = 2 X 70 - 25); one unit (and earn 70 = 70 - 0) or zero units (and earn zero). Without loss of generality, assume B1, B2 and B3 approach the sellers in rank order. Possible quantity outcomes for S1, along with their associated probabilities are shown in the bottom portion of Figure 5. Again, weighting the payoff for each outcome by the associated probability of occurrence, it is straightforward to show that S1's expected earnings from a unilateral deviation to a price of 70 are 65.185 cents. For any deviation smaller than seventy cents, per unit revenues fall without affecting sales probabilities. Thus expected earnings fall for any possible positive deviation.

APPENDIX C

The Uniqueness of "Trading" Nash Equilibrium at [P.sub.m] = 55 Under the Search Treatment

This appendix rules out asymmetric equilibria in which all three sellers have positive expected sales quantities. In all equilibria where some units trade, buyers expect that sellers will not price so high that shopping is unprofitable. Thus, in any equilibrium with trade, the expected price must be at least fifteen cents below the ninety-five-cent value of each buyer's first unit, or below eighty.

First consider the possibility of multiple symmetric equilibria. No common price between fifty-six and eighty cents can be an equilibrium, because each seller would have a unilateral incentive to raise price slightly. The price increase is not observed, so it does not affect the probability of sale. Similarly, at any common price below fifty-five cents, a small unilateral increase will not affect sales probabilities. Therefore, the only possibility is at fifty-five, which was shown to be a Nash equilibrium in appendix B.

Now consider asymmetric equilibria. No set of prices all below fifty-five cents could be an equilibrium, since the low-pricing seller would have a unilateral incentive to raise price. Similarly, there can be no asymmetric equilibrium with prices above fifty-five. This type of argument shows that the only remaining possibilities involve some combination of fifty-five and eighty cents, the highest price consistent with trade. If we assume that buyers know the equilibrium price strategies of each seller, then it is straightforward to show that either of the two sellers posting eighty cents would find a unilateral deviation to fifty-five cents profitable. Suppose S1 and S2 post eighty, and S3 posts fifty-five. In an asymmetric equilibrium the buyers know this and the first two buyers to shop purchase all four of the units available from S3. This leaves one remaining buyer who is willing to purchase only a single unit from either S1 or S2, who would each have a unilateral incentive to lower price to fifty-five cents where at least two units would be sold. Finally, a "no trade" equilibrium arises in the case where a single seller posts eighty and the other two sellers post fifty-five. In this case the buyers purchase six units from the two low-priced sellers, leaving the high-priced seller with no sales, independent of his or her price. Even though this is one of the equilibria we are excluding, note that it involves only trades at the monopoly price.

These kinds of arguments can also be used to rule out equilibria in mixed strategies. Since buyers do not observe prices before shopping, sellers cannot be indifferent over price ranges. Over the price range between twenty-five and fifty-five cents, for example, increases above twenty-five will increase revenue per unit without affecting sales probabilities. Randomization at prices over fifty-five can be ruled out in a similar way.

APPENDIX D

Results in Terms of Posted Prices

Tables III and IV present posted price information parallel to the transactions price information presented in the text. As is clear from the tables, the effect of the search treatment is somewhat weaker when presented in terms of posted prices. Nevertheless, using the nonparametric randomization test described in footnote 14, each of the affirmative conclusions drawn in the text are supported with posted price data at least at a 90 percent confidence level.

TABLE III

Mean Deviations of Posted Prices from the Competitive Prediction for the Final 5 Periods: Search and Posted Offer Treatments

 Search Sequence Posted-Offer Sequence
Session P-[P.sub.e] P-[P.sub.e]
 SP1 8 1
 SP2 26 3
 SP3x 25 2
AVG. 20 2
 PSI 17 19
 PS2 3 1
 PS3x 14 17
AVG. 15 12

TABLE IV

Mean Deviations of Posted Prices from the Competitive Prediction for the Final 5 Periods: Search with No Reputations

Search, No Reputations, Search, No Reputations,
 c = 15 c = 30
Session P-[P.sub.e] Session P-[P.sub.e]
SNL1 24 SNH1x 26
SNL2 7 SNH2x 32
SNL3x 15 SNH3x 23
AVG. 15 AVG. 27

Conclusion 1, for example, hinges on rejection of the null hypothesis that prices are no higher in the search sequences than in the posted-price sequences. Under this null hypothesis, the signing of the six differences observed in Table III are coincidental. Of the [2.sup.6] = 64 ways that signs for the six differences could have arisen, six generate a sum of differences as large or larger than that observed. This occurs with probability of 6/64 = .091. Similarly, conclusion 3 rests on the conclusion that prices in the SNH sessions are higher than in the SNL sessions. Given that we are talking about six independent deviations rather than six treatment effects, a different version of the randomization test must be applied. In this case, sum deviations for the three SNH sessions and for the three SNL sessions. Then difference the sums. Under the null hypothesis, the association of the observed deviations with particular treatments is coincidental, so the difference of the sums should cancel out. But of the [Mathematical Expression Omitted] = 20 possible ways that the observed deviations could have occurred, only two generate a total as large as that observed. This occurs with probability 2/20 = .10.

(1.) Much subsequent theoretical research focused on the sensitivity of this paradox to alterations in the information-transmission technology. The resulting models identify conditions under which monopoly, competitive and heterogeneous prices are predicted, depending on how information is disseminated. See, e.g., Butters [1977], Salop and Stiglitz [1977], and Wilde and Schwartz [1979]. (2.) Both intuition and the Diamond result would suggest that if goods are sold on a posted-price basis, then public information about those prices should improve performance. But it should not be assumed that the effect of public information is independent of other institutional aspects of the market. For example, Hong and Plott [1982] show that market performance is impaired, not improved, by the transition from unstructured bilateral negotiation (with no public price information) to posted prices (with public price information). In contrast, the Diamond paradox pertains to a comparison between costly and public price information, within a common institutional framework in which prices are posted on a take-it-or-leave-it basis. (3.) There is some argument to be made for providing traders with complete cost and value information when evaluating game-theoretic predictions, e.g., Davis and Holt [1994]. We did not provide such information here, for two reasons. First, we wished to evaluate search predictions in a market-like environment, where full cost and value information is typically not available. Second, as is clear from a review of ultimatum game experiments, summarized in Davis and Holt [1993, chapter 5], rivalistic behavior is typical in full information environments when earnings deviate significantly from an equal division of the available surplus. We wanted to keep such rivalistic behavior from preventing the monopoly or competitive outcomes predicted by Diamond-paradox arguments. (4.) Thus we compare performance across price-information conditions, holding seller-approach costs constant. This differs from the usual theoretical discussion, which focuses on the effects of injecting costly search into an environment where prices are not public. We chose not to evaluate performance in a nosearch cost/private-information baseline environment largely for procedural reasons: it takes each buyer several minutes to elicit costless price quotes from sellers each period. Not only would this treatment make it difficult to examine a pair of treatments in a two-hour session, but any observed differences in performance from that observed in a standard posted-offer market could be attributable to (uncontrolled) time-costs of shopping. (5.) Marginal cost and marginal value parameters are presented to participants via our posted-offer software. In trading, participants are not required to adhere to these constraints (that is, trades at a loss are permissible). However, prior to posting units at prices below costs, or making a purchase at an above-value price, the participant must acknowledge a warning that they are about to lose moneY In fact, there were very few trades at a loss, and those that did occur were confined to the initial periods. Finally, costs and valuations were all increased by $6.00. This has no effect on the equilibria and is omitted from the text for expositional ease. (6.) See, for example, chapter 4 of Davis and Holt [1993]. (7.) Verification that the joint-profit-maximizing price is fifty-five: At a common price of fifty-five cents, each seller can sell one low-cost and one high-cost unit, for earnings of eighty-five cents (= 2 X 55 - 25). At any common price below fifty-five, earnings are lower, since per unit revenues fall, and aggregate sales quantity cannot increase. At any price above fifty-five, sellers can expect to sell only one unit, and the highest price that will induce buyers to pay the approach fee is eighty cents. Thus, maximum earnings at any common price above fifty-five cents are eighty cents. (8.) There are also an infinite number of "no-trade" equilibria in which sellers price above eighty cents and buyers do not shop. These are weak Nash equilibria since a unilateral price reduction will not increase sellers' profits above zero. Most theoretical models avoid such equilibria by assuming that at least some buyers costlessly have access to some prices. It is typical, for example, to let buyers make their first approach free. We did not implement an assumption of this type here, as it is both rather artificial and is invoked to avoid a problem that is not behaviorally very important. With one exception (a high-search-cost session, described below), buyers always approached sellers. (9.) There are numerous other differences between our search treatment and that reported by Grether, Schwartz and Wilde. In addition to using the same participants for all sessions (as mentioned in the introduction) and posting the price distribution to buyers, these authors used thicker markets (five to eight sellers and twenty-five buyers); buyers each had a constant marginal valuation for a single unit; sellers had U-shaped cost curves (induced via a fixed cost and a capacity constraint); and sellers could not stock-out in a period. Rather, sellers were obligated to satisfy all realized demand even if they had to incur the fixed production costs again to start a new production run in the same period. (10.) Instructions for the baseline posted-offer treatment are printed in Appendix A4.2 of Davis and Holt [1993]. The additional instructions that were read aloud to participants to create the search treatment are available from the authors, on request. (11.) In retrospect, we might have announced the final period since we are evaluating a static prediction. Uncertainty about the final period would at least implicitly induce discounting and therefore might prompt price increases via the use of trigger strategies. The generally low prices in the posted-offer treatment suggest that tacit collusion was not a problem. (12.) All of the following analysis is presented in terms of transactions, rather than posted prices. Each of the conclusions drawn in the text is supported using posted prices, although at a slightly weaker level of confidence (but at least 90 percent). Relevant posted price information is presented in Appendix D. The slightly weaker effects of the search treatment on posted prices strengthens our overall conclusion that the absence of public price information does not quickly draw prices from the competitive to the monopoly prediction.

(13.) We confine our attention to the last five periods of each treatment sequence as a crude means of controlling for learning. (14.) The randomization test is an exact probability test, based on the differences in prices across treatments in each sequence. The test assumes independence of the treatment pairs and that both the magnitude and direction of price changes are meaningful. The relevant probability is calculated as the ratio of the number of ways that a sum of price differences at least as large as the observed sum could be generated, divided by the total number of possible outcomes. In the present case, this probability is very easy to calculate: since the search treatment raises prices in each session, the observed sum of price differences is the largest possible out of the [2.sup.6] = 64 possible outcomes. Under the null hypothesis of no treatment effect, an outcome this extreme would be observed 1/64 (= .015) of the time. For a more complete description of the test see Conover [1980]. (15.) The heterogeneity of outcomes both within and across treatments also indicates that factors other than the presence of public price information may affect pricing behavior. Some of these factors are largely procedural. In particular, an order-of-treatment effect is suggested by the mean price information in Table I. Average deviations are higher in sequences where the search sequence occurred first (eighteen cents vs. eleven cents), but the difference is not significant at conventionally accepted confidence levels. (Using the randomization test, the null hypothesis of no order-of-sequence effect can be rejected at only an 80 percent confidence level.) (16.) Although this treatment may be expected to shed some light on the importance of seller reputations on pricing outcomes, it should not be interpreted as a test of the Bagwell-Ramey model. An exact implementation of the Bagwell-Ramey model would require too many design alterations to allow comparison of results with our existing sessions. In particular, buyers and sellers should have full information about values and costs. Moreover, although indefinite repetition is not necessary to generate higher prices in the search sessions, either the final period, or announcement of a termination rule is necessary to give the theoretical predictions a reasonable chance of being observed. (17.) Actually, we decided to focus on the effects of reputation in the initial session after observing almost perfectly competitive prices in the search sequence of a no-reputation pilot session where the search sequence followed a posted-offer sequence. The reported sequences were followed with a variety of pilot treatments, none of which are particularly relevant to this discussion. (18.) There were two exceptions. First, our experience profile consisted entirely of experienced participants. Second, buyer earnings were supplemented with a private, one-time $5 payment in the middle of each session. We were prompted to provide this supplement while observing very low buyer earnings in session SNH1x, which threatened to diminish buyer interest. (19.) Intuitively, of the [Mathematical Expression Omitted] = 20 possible ways that the average price deviations for the six sessions in the SNL and SNH treatments could be arranged into groups of three, the observed outcome generates the largest possible difference between the sum of the group deviations. Under the null hypothesis of no treatment effect, this would occur one time out of twenty. (20.) Other differences across our design and the Grether, Schwartz and Wilde design are important, but are clearly less likely candidates for motivating price increases. In particular, in each of the Grether Schwartz and Wilde sessions, there were either five or eight sellers, compared to the three sellers used here in each session. Indeed, ex ante a critic might argue that it is our design that is biased toward monopoly outcomes, and it may well be the case that increasing the number of sellers in our design will generate yet more competitive outcomes. Nevertheless, the high prices observed with a relatively large number of sellers by Grether, Schwartz and Wilde, combined with the more competitive outcomes we observe in three-seller markets, suggests that standard pure-structure effects are dominated by other factors when price information is not public. (21.) Klemperer [1992] provides a nice summary of issues in the switching cost literature.

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CHARLES A. HOLT, Virginia Commonwealth University and University of Virginia, respectively. We thank without implicating two anonymous referees and Kyle Bagwell. This research was supported by grants from the University of Virginia Bamkard Fund and the National Science Foundation (grants SBR 9319842 and SBR 9320044). Data reported in this paper are available at FTP address: fido.econlab.arizona.edu.