Examine the pricing strategies in the gasoline market. Make sure to address the following topics:

In the article, (Mixed) Strategy in Oligopoly Pricing: Evidence from Gasoline Price Cycles Before and Under a Timing Regulation, the author's final conclusion is that "the price leadership outcome under the law is better predicted by mixed strategies play than by alternative hypotheses." Did the author convince you that "the price leadership outcome under the law is better predicted by mixed strategies play than by alternative hypotheses"? Why or why not?

1. Do you think there should be any policy response to the tacit pricing coordination documented in this paper? Why or why not?
2. The author states that "since price rigidity exists in many industries, the results suggest that the idea of reaction based on short-run price commitment may be applicable in many other settings as well." Identify at least one other setting in which this approach would work well. Why do you think it would work? Conversely, identify one setting where it would not be successful. Explain.

Article:

(Mixed) Strategy in Oligopoly Pricing: Evidence from Gasoline Price Cycles Before and Under a Timing Regulation Zhongmin Wang Northeastern University This paper studies oligopoly firms' dynamic pricing strategies in a gasoline market before and after the introduction of a unique law that constrains firms to set price simultaneously and only once per day. The observed gasoline pricing behavior, both before and under the law, is well captured by the Edgeworth price cycle equilibrium in the Maskin and Tirole dynamic oligopoly model. My results highlight the importance of price commitment in tacit collusion. I also find evidence that the price leadership outcome under the law is better predicted by mixed strategies play than by alternative hypotheses. I. Introduction This paper examines the oligopoly problem by studying firms' dynamic pricing strategies in a retail gasoline market before and after the intro- duction of a unique law that constrains firms to set price simultaneously and only once per day. The observed gasoline pricing behavior, both before I thank the editor and two anonymous referees for their comments, which significantly improved this paper. I am especially grateful to Jim Dana for many helpful and stimulating discussions. I thank Kamran Dadkhah, Juan Dubra, John Kwoka, Christian Rojas, and seminar participants at Clark University for their comments on an earlier version of this paper. I also thank Simon Goss, Sean Isakower, Shu Li, and Jingjing Yang for their assistance with collecting data; Allan Price at Informed Sources; staff members at the Western Aus- tralia Department of Consumer and Employment Protection; the owners of three gasoline retailers for providing data; several industry participants for explaining the Perth gasoline market and providing evidence on the special forms of vertical restraints documented in this paper; and the Faculty of Business and Economics at Monash University for providing financial support. Any errors are mine only. [ Journal of Political Economy, 2009, vol. 117, no. 6] @ 2009 by The University of Chicago. All rights reserved. 0022-3808/2009/11706-0001$10.00938 JOURNAL or Pou'rrcsr. ECONOMY and under the law, is well characterized by the Edgeworth price cycle equilibrium in the Maskin and Tirole (1938) model that features short- run price commitment. My results thus highlight empirically the im- portance of price commitment in tacit collusion. I also nd evidence that the price leadership outcome under the law is better predicted by mixed strategies play than by alternative hypotheses even though rms have the incentive not to deliberately randomize. Central to the oligopoly problem is the question of how price is formed in oligopoly markets. Given a market setting, strategy describes behavior. The question then becomes what strategies oligopoly rms use to form price. Gametheoretic oligopoly theories have mainly con- sidered two classes of dynamic strategies. The MaskinTirole approach presumes that once a price is set, it cannot be changed in the short run, because of price rigidity or commitment, so that another rm can subsequently react to that price. Thus, a rm reacts to a past price because that price aects its payo'. In contrast, in the canonical re- peated game or supergame model, a Bertrand game is independently repeated so that past prices have no tangible effect on the current market: nothing prevents rms from changing price ever period. Hence, a rm elects to condition its current price on past prices only because it decides to do so, not because past prices directly aect its Payoff- There are many game-theoretic models of tacit collusion in the lit- erature (see Tirole [1988], Shapiro [1989], and Viva [2001] for ref- erences), but there is relatively little empirical evidence on oligopoly rms' actual dynamic pricing strategies in real markets.1 A major reason for this imbalance between theory and evidence is that much of the discussion on tacit collusion is based on the supergame approach, but the strategies used in supergame models are typically not intended as descriptions of what rms actually do in reality. For example, the folk theorem is a key result of the supergame approach, yet according to Mailath and Samuelson (2006, 73), it \"says nothing about behavior. The strategy proles used in proving folk theorems are chosen for analytical ease, not for any putative content, and make no claims to descriptions of what players are likely to do.\" This paper presents evidence that the strategies featured in the MaskinTirole model eectively describe rms' dynamic pricing behavior in a real market. This paper's window into rms' pricing strategies is a unique lawr that regulates the timing and rm of price setting. The law, a remarkable market experiment, took effect in Western Australia inJanuary 2001. It 1There is a literature on how cartels work (see Gene-sow and Mullin [2001] for an excellent example). My focus is on tacit collusion. There is also anecdotal evidence on facilitating practices such as price leadership. I analyze price leadership as part of rms' dynamic pricing strategy. (MIXED) STRATEGY IN OLIGOPOLY PRICING 989 requires every gasoline station in the Perth metropolitan area to (1) notify the government of its next day's retail prices by 2:00 p.m. each day so that these prices can be published on an Internet Web site, and (2) post the published prices on its price board at the start of the next day for a duration of at least 24 hours. This law thus forces firms to set gasoline prices simultaneously (without knowing rivals' prices) and at most once every 24 hours. In this paper, I document firms' pricing strategies before and under the law using a rich and high-frequency data set that tracks the price changes of nearly every gasoline site in the Perth area and the daily changes in marginal costs of supplying wholesale and retail gasoline. Figure 1 shows the hourly (from 6:00 a.m. to 6:00 p.m.) brand average retail gasoline prices of three firms in the Perth area for a period of 39 days before the law. Figure 2 shows the daily brand average retail prices of three firms and a wholesale gasoline price series for a period of 57 days under the law. Figure 3 shows a theoretical example of Maskin and Tirole's Edgeworth price cycle equilibrium. In all three figures, firms hike price sequentially and then decrease price gradually, and at the bottom of each cycle, a war of attrition problem arises. Price increases by all would benefit all, but none would like to be the first to hike price. My empirical evidence supports Maskin and Tirole's theory that price commitment is important to tacit collusion. Before the law, a lead price hike would stick for a couple of hours; it is simply too costly to change price every hour or minute, even in the gasoline market. Thus, rival firms can observe and react to the lead price hike within hours before the law. On the other hand, a firm must be committed to its price increase for at least a day under the law. This large increase in the length of price commitment implies a much higher cost for price lead- ership: a leader has to lose market share for an entire day before any rival firm can respond. Consistent with this change, I find that a single large firm was nearly always the first to hike price before the law, but price leadership under the law is distributed among the three largest firms. Mixed strategy is part of the dynamic strategies used by Maskin and Tirole to derive the Edgeworth price cycle equilibrium. Specifically, they presume that firms play a mixed strategy to decide price leadership at the bottom of each price cycle. In this paper, I take this assumption seriously. Mixed strategy is a fundamental concept in game theory, widely used in both zero-sum and nonzero-sum games. Because naturally occurring Price leadership is "one of the most important institutions facilitating tacitly collusive pricing behavior" (Scherer and Ross 1990, 346). The Maskin-Tirole model generates price leadership as part of the price cycle equilibrium.Cabar Mobil Price: Australian cents per Iner 990 07/11/00 1fam 07/18/00 11am 07/25/00 11am 08/02/00 11am 08/08/00 11am 07/04/00 11am FIG. 1.-Hourly brand average gasoline prices over six cycles before the lawPrice: Australian conis per liter 991 Capex Shell Wholesale price 08/27/02 09/03/02 09/10/02 09/17/02 09/24/02 10/01/02 10/08/02 10/15/02 10/22/02 FIG. 2.-Daily brand average gasoline prices over seven cycles under the law\f(MIXED) STRATEGY IN omens-our PRICING 993 strategic situations are often too complicated to be suitable settings for empirical testing, nonexperimental tests of the mixed strategy concept have been limited to the zero-sum games of tennis serves and soccer penalty kicks (Walker and Wooders 2001; Chiappori, Levitt, and Grose- close 2002; PalaciosHuerta 2003; Hsu, Huang, and Tang 52007).3 The use of mixed strategy in zero-sum games is relatively intuitive since play- ers in such games have the incentive to deliberately randomize to remain unpredictable (von Neurnarm and Morgenstern 1944, 145). However, the use of mixed strategies in nonzero-sum games is rather counterin- tuitive since players in such games often have the incentive to avoid randomization (Schelling 1960, 175) . This raises the question of whether there are any empirically tractable strategic situations in which players do not have the incentive to randomize, but their actions are well char- acterized by equilibrium mixed strategies. This is the rst paper in the literature to study such a strategic sitti- ation: the war of attrition game at the bottom of the gasoline price cycles. Firms in this game have the incentive to avoid randomization since pure strategies would end a war of attrition immediately without incurring any cost of delay. In this paper, I can test the mixed strategy hypothesis using techniques similar to those used to test mixed strategies in sports games. First, the actions in this game are discrete (either relent or ght), and the outcomes are the identities of the price leader's. Sec- ond, the same game is repeated many times, providing rich variations in the observed outcome. The rms did not play mixed strategies before the law when a single large rm appeared to serve as the market leader. However, I nd em- pirical evidence for mixing behavior under the law. First, the leadership outcome of the individual wars of attrition under the law, once condi- tional on the outcome of the previous war, is random. Second, the stochastic regularities of the leadership outcomes (leadership types and their frequencies) are captured reasonably well by the mixed strategy presumed by Maskin and Tirole. The observed mixing behavior differs from that presumed by them in one important aspect: The outcomes of the wars of attrition are serially correlated, suggesting that the rms may have attempted to coordinate over the wars of attrition even under the law. The nding of mixing behavior under but not before the law is consistent with Harsanyi's (1973) Bayesianjustication for mixed strat- egy: A player's mixed strategy represents other players' uncertainty of 5 The results from experimental tests are rather mixed (see, e.g., Walker and Wanders [2001] for a review). A literature estimates game-theore tics] models that may involve mixed strategies (13.32, Hendricks and Porter 1988). 994 JOURNAL OF POLITICAL ECONOMY that player's pure choice.* Under the law, firms are uncertain about rivals' actions when deciding whether to hike price, but that uncertainty does not exist before the law. A regular gasoline price cycle is not unique to the Perth market. It appeared in many U.S. cities in the 1960s (Castanias and Johnson 1993), and it is currently occurring in many U.S. cities in the Midwest (Lewis 2009) and in Canada (e.g., Eckert and West 2004; Noel 2007). Without observing a market experiment or being free from data constraints, these studies do not highlight the importance of short-run price com- mitment or test the mixed strategy hypothesis. The rest of the paper proceeds as follows. Section II discusses Maskin and Tirole's model of dynamic oligopoly pricing. Section III describes the market setting and the data set. Section IV documents that key regularities of firms' pricing behavior are characterized by Maskin and Tirole's Edgeworth price cycle equilibrium. Section V tests the mixed strategy hypothesis under the law. Section VI evaluates the welfare im- pact of the law, and Section VII presents conclusions. II. Maskin and Tirole's Model of Dynamic Oligopoly Pricing The idea of reaction based on commitment traces back at least to the classic Stackelberg model. Ever since Schelling (1960), commitment has been recognized as a central feature of much strategic behavior. Maskin and Tirole (1988) formalize the idea of reaction based on commitment in a fully dynamic setting. To capture the existence of short-run price commitment, they assume that a price, once set, lasts for two periods. One justification for this assumption is that price is rigid in the short run because of exogenous price adjustment costs. Price rigidity thus can serve as a commitment device. To capture the idea of reaction, Maskin and Tirole presume that two firms set price alternatingly when com- peting repeatedly over the price of a homogeneous good. Firms follow Markov reaction strategies in that a firm responds only to the price set by the rival firm in the previous period, which is the only payoff-relevant variable. Either the Edgeworth price cycle equilibrium or the kinked demand curve equilibrium arises as a Markov perfect Nash equilibrium in this model. Since equilibrium profit in either case is well above the one-shot Bertrand profit, tacit collusion is not only possible but nec- essary in this model. Harsanyi (1973) shows that almost any mixed strategy equilibrium can be viewed as a pure strategy Bayesian equilibrium in a nearby game in which the payoffs to each player are subject to small private random variations. Aumann (1987) takes Harsanyi's idea further and directly interprets a player's mixed strategy as an expression of other players' uncertainty of that player's choice of pure strategy. See Reny and Robson (2004) for a recent discussion of the interpretation of mixed strategy.(MIXED) STRATEGY IN OLIGOPOLY PRICING 995 Recall figure 3 for an example of the Edgeworth price cycle equilib rium, which has two rare features. First, equilibrium prices change over time even though the underlying demand and cost remain constant, and second, equilibrium prices are directly indicative of the underlying pricing strategies. If price is at marginal cost, firms are in the war of attrition phase: Both firms want to hike price, but neither wants to be the first to do so because the leader loses market share whereas the follower gets a free ride. For the cycle equilibrium to arise, however, the public good of price leadership must be provided. Maskin and Tirole presume that firms play a mixed strategy to decide price leadership. This technical assumption is elaborated immediately below. Once a firm relents by hiking price, the other firm reacts with a slightly smaller increase in the following period. These two price increases constitute the rising phase of a price cycle. In the subsequent falling phase, the two firms undercut each other gradually until price reaches marginal cost. Maskin and Tirole presume that firms decide price leadership by playing the standard stationary mixed strategies in wars of attrition: firm i always relents with probability p, in period t conditional on no firm having relented before then. When a rival's price is at the competitive level, a firm's best response is to attach probability p, to relent (i.e., increase price) and 1 - p, to fight (i.e., keep price at marginal cost). In any period of an attrition war, a firm is indifferent between relenting and fighting, with the value of an action given by a Bellman equation. Mixed strategy is a convenient technical device here. By playing a mixed strategy, both firms have a chance to be the leader for any cycle so that the burden of price leadership is shared over time. In the prelaw Perth market, however, firms were unlikely to play mixed strategies. Players in wars of attrition have the incentive not to random- ize: mixed strategies lead to costs of delay whereas pure strategy equi- libria end a war of attrition game immediately. In addition, the disin- centive to be a leader before the law is quite small. The gasoline firms constantly monitor each other's price changes, so once a firm relents, rival firms can quickly follow by hiking their price as well. This reasoning suggests that a single firm may be willing to serve as the price leader. The law does not change the basic intuition behind the price cycle equilibrium, which, in the words of Tirole (1988, 256), is "that if firms were stuck in the competitive price region, with the prospects of small profits, a firm could raise its price dramatically and lure its rival to charge a high price for at least some time (the rival would not hurry back to nearly competitive prices)." Indeed, most postlaw changes in the ob- served gasoline price cycles are intuitive to the point of being self- explanatory. For this reason, I discuss only briefly the impact of the law on the war of attrition game and on welfare. The timing law changes the war of attrition game at the bottom of996 JOURNAL OF POLITICAL ECONOMY each price cycle in significant ways. As mentioned earlier, the cost of providing price leadership is much greater now; the leader must lose market share for at least 24 hours before any rival firm can respond. This implies that price leadership needs to be allocated among the firms. Indeed, postlaw price leadership in Perth is distributed among the three largest firms. Under the law, firms still have the incentive to coordinate on pure strategy equilibria. To do that, however, the three largest firms must be certain about each other's pure action at the beginning of each war, which is implausible under the law. Firms cannot credibly signal their intent at the bottom of the price cycles under the law. The timing of the wars of attrition game is thus forced to be discrete and simul- taneous. Consequently, on a day of a war of attrition, a firm may decide to relent or fight, but rival firms are uncertain about that firm's pure action because firm-specific private information always exists. Therefore, the presumed mixed strategies may describe the leadership pattern un- der the law. In the short run, the welfare impact of the law is clear. It lowers average retail gasoline price because it disrupted the gasoline firms' pricing coordination. Section IV.B.1 documents that regular gasoline price cy- cles disappeared after the law took effect (see App. fig. Al). However, after the firms succeeded in coordinating on the price cycle equilibrium under the law, it is not clear whether the average retail price is higher or lower than it was before the law. III. Market Setting and Data Set The Perth gasoline market resembles Maskin and Tirole's (1988) model environment. Only a few firms are in the market during the sample period. Price is the primary strategic variable, and retail gasoline price is publicly observable. Gasoline is a relatively homogeneous product," and the demand for gasoline is stable on a daily basis." The only sig- nificant discrepancy is that the cost of gasoline varies significantly over time as oil price fluctuates. A major feature of the market setting is that it allows one to construct cost measures that closely track the daily changes in marginal costs of supplying wholesale and retail gasoline. During the sample period July 1, 2000, through October 31, 2003, the major gasoline firms in the Perth market include four oil firms (BP, Wang (2009) collects daily station-specific gasoline sales data in the Perth market and uses the timing of the regular price cycles as instruments to estimate station-level gasoline demand. His elasticity estimates confirm that drivers in the Perth market are highly price sensitive to station-level gasoline price differentials: the estimated own price elasticity is as large as -18.8. "In any case, changes in gasoline demand have little effect on the price cycle dynamics. Gasoline firms face the same demand shocks, yet it is often the case that some firms increase price on a day but others do not.(MIXED) STRATEGY IN OLIGOPOLY PRICING 997 Caltex, Shell, and Mobil) and two independent firms (Gull and Peak). BP operates the only refinery in Western Australia. Before July 2002, Caltex, Shell, and Mobil all obtained fuel from BP through reciprocal refinery exchange programs. Since July 2002, Caltex and Shell have been buying fuel from BP, and Mobil has been importing fuel from Singapore. During the sample period, Gull bought fuel from BP and Peak bought fuel from Mobil. Caltex and Shell buy fuel from BP through contracts that are typically renewed every 6 months. It is important to note that BP's gasoline pricing is constrained by potential import from Singapore because the BP refinery is "small in scale and less efficient than refineries in the Asia-Pacific region, particularly the large modern refineries in Singa- pore" (Australian Competition and Consumer Commission 2007, 100). For this reason, Caltex's and Shell's purchase contracts specify that the price that they pay BP is determined by a formula tracking the potential cost of importing from Singapore. This feature of the market allows one to construct a cost measure that closely tracks the movements in Caltex's and Shell's marginal costs of supplying wholesale gasoline. By law, oil firms in Australia can own or operate only a small number of retail sites.' It is widely acknowledged, however, that the law is inef- fective in preventing oil firms from controlling retail gasoline price. BP, Shell, and Mobil used what are called multisite franchise agreements to control their retail gasoline price." For example, Shell had a single franchisee that essentially operated all the Shell branded retail sites in the Perth market, and a report by the WA government concluded un- equivocally that BP directly controlled the retail prices of its multisite franchisees (Western Australia Select Committee on Pricing of Petro- leum Products 2000, 38)." Prohibited from using multisite franchise agreements, Caltex used a different mechanism, called the conditional price support, to effectively control the retail price of its many fran- chisees. Appendix B documents this interesting form of vertical restraint. I describe below the retail price and cost indicators used in this paper. The price data used to evaluate the welfare impact of the timing law are described in Section VI. The unit of all price and cost data through- out the paper is Australian cents per liter. The law is called the Petroleum Retail Marketing Sites Act 1980 (the Sites Act). Ac- cording to the Sites Act returns for March 2002, BP owned six sites, Caltex owned 22 sites, and Mobil and Shell owned no sites in the Perth market. "The Sites Act was repealed in 2007, perhaps an admission of its ineffectiveness. "The multisite franchise agreements are known to be consignment agreements that allow the oil firms to retain the ownership of fuel until it is sold at retail, thus giving the oil firms the right to set retail prices. "The multisite franchise agreement "exhibited characteristics more consistent with commission agency than franchise operations."998 JOURNAL or POIJTIGAL ECONOMY A. Retail Price To study the price cycles under the law,11 I use a panel data set that records the daily (regular unleaded gasoline) price of all retail sites in the Perth market from the start of the law on January 3, 2001, through October 31, 2003.12 During this period, BP, Caltex, Shell, and Mobil operated or controlled an average of 67, 88, 45, and 23 sites per day, respectively. Gull and Peak operated an average of 33 and 13 sites per day. These six brands accounted for about 85 percent of the retail sites in the Perth metropolitan area. Appendix gure C1 shows the daily market average price from the start of the law through October 31, 2003. To study the price cycles before the law, I use a panel data set that covers the retail price of nearly every gasoline site in the Perth area for the periodJuly 1 to December 20, 2000.13 For three brands (BP, Caltex, and Mobil), the price data are hourlybetween 5:00 am. and 5:00 p.m. each day, 7 days a week. The hourly prices, critical to the identication of price leaders, were electronically sourced from purchase transactions with gasoline credit cards. 1\" For Shell and the independent brands, retail prices were collected (via drive-by) twice a daybetween 5:00 a.m. and 9:00 a.m., and noon and 3:00 p.m., Monday through Friday. The number of gasoline sites covered in the data set before the law is slightly smaller than that under the law. For example, on July 1, 2000, the prelaw data set covers 285 stations: 89 Caltex sites, 73 BP, 35 Shell, 25 Mobil, 30 Gull, 7 Peak, and 25 sites of several small independent brands. B. Cast Indicators Since the marginal cost of supplying gasoline varies considerably even in the short run, it is important to study the potential impact of cost changes on pricing dynamics. To do so, I use two types of cost measures that capture the short-run variations in suppliers' marginal costs of sup- \" The timing law is called the 24-hour rule. The Internet Web site established by the law mentions that \"motorists' frustration at in Ira-day price uctuations and the signicant di'erence between city and country fuel prices\" were the political reasons that led to the 24-hour rule. There was a loophole to the law prior to August 24, 2001. During that period, a station must mimits next-day retail price butis not required to move to the nominated price. Since the major rms in the market did not: take advantage of this loophole, it does not alfect the analst in this pap-e1: 1' The data were downloaded from the Internet Web site (http://www.i'uelwatnchmra .govau) established by the law. 1' The data were collected by Informed Sources, a market research rm in Australia 1" Many drivers in Australia purchase gasoline using gasoline credit cards (e.g., Caltex Star Card). Each time a gasoline credit card is used to purchase gasoline at a retail site, the retail price at the pump is sent electronically to Informed Sources. The price data provided to the author are the latest prices for each station at each hour between 6:00 a.n1. and 5:00 p.m. (MIXED) STRATEGY IN OLIGOPOLY PRICING ggg plying gasoline: the condential wholesale transaction prices paid by three retailers and a cost measure that closely tracks the movements of the actual gasoline price that Caltex and Shell pay BP. Other cost com- ponents, such as labor, inventory, and storage, are presumably xed in the short run. The three retailers are a BP franchisee that owns several sites and sets retail price independently, a small independent retailer,15 and a major independent retailer.\"5 The wholesale price shown in gure 2 is the wholesale price paid by the small independent retailer. I am able to estimate the gasoline price that Caltex and Shell pay BP because it is determined by a pricing formula that tracks the potential cost of im- porting gasoline from Singapore. The pricing formula, called the import parity pricing ([PP) formula, is the following: [PP-based import cost = a benchmark gasoline wholesale price in Singapore + shipping cost + quality premium + wharfage + insurance and loss + tax. Detailed explanations of this formula can be found in Western Aus- tralia Department of Consumer and Employment Protection (2007, 19 22) or Australian Competition and Consumer Commission (2007, chap. 7}. The Platts quote for the gasoline specication of Mogas 95 is the Singapore benchmark wholesale gasoline price, which drives the vast majority of the variations in the import cost. I have access to the daily IPP-based import cost for the 10-month period January 1 through Oc- tober 31, 2003.1\" am not able to estimate BP's marginal cost of rening 1" This retailer initially bought fuel from Shell but later changed its fuel supplier to some other rm. 1' The major independent retailer's wholesale buying price is available for the period January 1, 2001, through june 30, 2002. The wholesale buying prices paid by the BP franchisee and the small independent retailer, respectively, are available for the period January 1, 2001, through October 31, 2003. The BF franchisee's wholesale buying price is missing for three months. 17 The import cost data were obtained from the Western Australia Department of Con- sumer and Employment Protection. The department did not provide the import cost series directly. Instead, the department provided the wholesale margin series, which is dened as the diEerenee between the average terminal gate price and the import cost By a regulation similar to the 24-hour ruleI oil rms in Western Australia must report to the government by 2:00 p.m. each day their next day's terminal gate price (which is the maximum wholesale price an oil rm can charge any retailer). Since the terminal gate prices are published on the Internet, the import cost series can be calculated. 1000 JOURNAL OF POLITICAL ECONOMY regular unleaded gasoline," but this does not affect the analyses sig- nificantly." Figure 4 shows Caltex's and Shell's brand average retail price, two retailers' wholesale buying price, and the import cost for the period July 20 through October 20, 2003. The import cost explains over 99 percent of the variations in the level of the BP franchisee's wholesale price, and the first difference in import cost explains 29 percent of the first difference in the BP franchisee's wholesale price. The regular cycle in retail price is clearly absent from the cost series. Indeed, the first difference in import cost explains 0.00 percent of the first difference in the daily retail price of either Caltex or Shell. To emphasize this point, figure 5 plots the corresponding daily margins for Caltex and Shell, defined as the difference between brand average retail price and the import cost. The margin series exhibit a cycle that is essentially identical to the retail price cycle except that the wholesale price trend is removed from the margin series. Moreover, import cost decreased monotonically, with a single exception, over the 27-day period Septem- ber 4 through September 30, 2003, yet retail price continued to cycle during this period. Hence, cost changes cannot explain the existence of the price cycles. Nonetheless, the logic of the Maskin-Tirole model implies that cost changes may affect the price cycles in subtle ways, which I document in Section IV.C. IV. Pricing Dynamics Figures 1-5 suggest that the basic patterns of the gasoline price cycles, both before and under the law, are well captured by the Edgeworth price cycle equilibrium. The gasoline firms hike prices sequentially, de- crease prices gradually, and confront the issue of which firm will be the first to increase price at the bottom of each cycle. This section uses the rich data set to document the gasoline firms' pricing strategies. In par- ticular, I highlight the critical role of short-run price commitments in generating the regular price cycles. Subsection A documents the pricing behavior before the law. Subsection B describes how the firms adapted " It is difficult to estimate accurately a refiner's daily marginal cost of refining regular unleaded gasoline. First, the actual crude oil price paid by BP and other components of BP's production costs are not publicly available. Second, BP's refinery in the Perth area refines several types of petroleum products simultaneously, so it is difficult to determine the cost of a particular refined product. "The import cost should also track the movement in the actual price Gull pays to BP and Mobil's actual cost of importing. Gull has import facilities, so it pays a price that tracks the potential cost of importing as well. The Australian Competition and Consumer Com- mission (2007, 208) suggests that, as an independent gasoline chain, Gull's potential import cost is slightly higher than that of Caltex and Shell. The report also suggests that Mobil's actual cost of importing is not substantially different from the IPP-based import cost.100 95 06 85 80 07/23/03 08/13/03 09/03/03 09/24/03 10/15/03 Caltex brand average retail price Shell brand average retail price ----- Wholesale price a BP franchisee pays ..- Wholesale price a small retailer pays IPP-based Import cost FIG. 4.-Daily brand average retail price, wholesale price, and import cost, July 20 to October 20, 2003\f(MIXED) STRATEGY IN OLIGOPOLY PRICING 1003 to the law and compares the pricing behavior before and under the law. Subsection C studies the impact of cost changes on the price cycles. A. Pricing Behavior before the Law 1. Intrabrand Synchronization and Uniformity in Price Hikes Figures 1, 2, and 4 indicate that the decision makers in the Perth market are the small number of oil and independent gasoline firms, not the hundreds of retail gasoline sites. The price hikes exhibit a conspicuous sequential pattern by brand, a pattern that can arise only if there is strong intrabrand synchronization. Indeed, the gasoline firms synchronize and homogenize intrabrand price hikes to facilitate interbrand pricing coordination. To see intrabrand synchronization and uniformity directly, consider the rising phase of the second price cycle shown in figure 1, which took place on July 13, 2000. The average BP price increased from 87.2 cents per liter at 11:00 a.m. to 88.9 cents at noon and then to 91.9 cents at 1:00 p.m. The average increase from 87.2 to 88.9 cents occurred because the prices at 13 BP sites were increased from 86.5 to 92.9 cents between 11:00 a.m. and noon whereas the other 60 BP sites' prices were not changed. The average increase to 91.9 cents occurred because the prices at another 35 BP sites were increased to 92.9 cents (mostly from 86.5 cents) between noon and 1:00 p.m. Thus, between 11:00 a.m. and 1:00 p.m., 48 of 73 BP sites hiked price to exactly 92.9 cents. Since no sites from other brands increased prices by 1:00 p.m. that day, we observe strong (albeit not perfect) intrabrand synchronization and uniformity in price hikes.20 Intrabrand synchronization and uniformity in price increases are im- portant to interbrand pricing coordination. By hiking the price of a large number of retail sites to a uniform level, a price leader makes a credible commitment, thereby inducing rival firms to feel the effect of strategic complementarity and establishing a single target at which its rivals will aim. Through intrabrand synchronization and uniformity, rival firms can react quickly to the first price hike, and the price leader can easily verify if its price increase has been matched in a timely manner. 2. Price Leadership and Followership There are 21 regular price cycles for the prelaw sample period. BP was the first to hike price in 18 of the 21 cycles, and Shell was the leader for the other three cycles. In those three cases, BP started to increase price within an hour. Caltex, the largest firm in the market, was never " Price decreases typically do not exhibit strong, if any, intrabrand synchronization or uniformity.1004 JOURNAL OF POLITICAL ECONOMY the rst to increase price before the timing law. Since Caltex served as a leader most often under the law, I do not consider this prelaw price leadership pattern to be consistent with mixed strategy play. This lead- ership pattern probably reects BP's position as the market leader since it owns the only renery in the Perth area. This pattern also reects the fact that BP's major rivals typically followed BP's initial price hike very quickly and that BP temporarily retracted its price hike if a major rival did not quickly follow. The rst price hike for each of the 21 cycles always occurred between 11:00 am. and 2:00 p.m. on Tuesday (ve cycles), Wednesday (eight cycles), or Thursday (eight cycles). The length of a cycle, dened as the period between two lead price hikes, was 6 days (eight cycles), 7 days (seven cycles), 8 days (three cycles), or 9 days (three cycles). Caltex typically followed within 2 or 3 hours and Shell within 3 or 4 hours. The amount of time it took Mobil, Gull, and Peak to follow suit tends to be less precise, but for the vast majority of the 21 cycles, Mobil followed by 5:00 a.m. of the second day, and Gull and Peak followed by 9:00 a.m. of the second day.21 The oil rms tend to increase price to match the price leader whereas the independent rms tend to slightly undercut the price leader (typically by 0.2 cent). For example, on July 13, 2000, after BP increased most of its sites' prices to 92.9 cents per liter, most Caltex, Shell, and Mobil sites matched this price, but most Gull and Peak sites increased prices to only 92.7 cents. 3. Price Re traction BP retracted its price increase temporarily if either Caltex or Shell did not follow quickly. Price retractions, visible in gure 2, may be partial or full. For instance, between 4:00 p.m. and 5:00 p.m. on july 13, 2000, 15 of the 55 BP sites that had hiked prices retracted those hikes and returned to their previous price levels. BP's price retractions over six other cycles are much more pronounced as many more BP sites retracted price hikes, and in two of these six cases, all sites returned to their prehike levels. Price retraction is important to our rmderstanding of the rms' pric- ing strategy. Price retractions are part of the Edgewtorth price cycle equilibrium if the MaskinTirole model has three or more rms (Noel 9' For about half of the cycles, Mobil followed the leader within a few hours, but for the other cycles, the data are such that Mobil started to hike price at 6:00 am. the next day. Because the data between 6:00 p.m. and 6:00 am. are not available, itis possible that Mobil may have started to hike price before 6:00 am. In the data, Grill and Peak always started to hike price on the second day, mostly between 7:00 am. and 9:00 am. Because the data for Gull and Peak were collected only twice a day, it is possible that these two brands may have sometimes started to hike price earlier than what the data indicate. (MIXED) STRATEGY IN OLIGOPOLY PRICING 1005 2008). In such a model, after one rm relents by increasing its price, the two remaining rms may not follow immediately because they still have the incentive to be the last to increase price. Hence, the leader may temporarily retract its price increase. Price retraction highlights that the lead price hike represents a short-run price commitment. Tem- porary price retraction also suggests that BP's price commitment is partly endogenous. BP's lead price hike is a commitment partly because it involves a loss of market share that carmot be recovered, not purely because BP cannot change its price quickly. B. Biting Behavior under the Low 1. Short-Run Price Commitment during the Adjustment Period Short-run price commitments emerged quickly after the law took effect, adding to the evidence that short-run price commitment is central to the gasoline rms' pricing behavior. Appendix gure A1 indicates that regular gasoline price cycles disappeared after the law took effect on January 3, 2001, and were not reestablished until early May 2001. How- ever, Shell started to initiate large price increases as early as February 12, and Caltex followed suit in early March. Furthermore, it is important to note that both Shell and Caltex kept their price hikes on the second day, a display of price commitment beyond a single day. Shell retracted its price hike on the third day if either BP or Caltex did not fully follow, and Caltex did the sanre if either B? or Shell did not fully follow. Thus, it appears that short-run price commitments and subsequent price re tractions were used to commtmicate intent and to help coordinate the new price cycles. BP did not initiate a price hike until April 18, and when it did, both Caltex and Shell followed on the second day, and a new regular price cycle started to emerge. 2. Pricing Dynamics after the Adjustment Period There are 102 regular gasoline price cycles from May 10, 2001, through October 21, 2003. Intr'abrand synchronization in price hikes is stronger in these cycles because of the need to increase price quickly. Before the law, a rm could hike some of its sites' prices one hour and others the following hours; a similar staggering of increases would take days to implement under the law. Once the new price cycle is established, lead price hikes are rarely retracted. This is expected since temporary re- traction that lasts a few hours carmot exist under the law. There are three cases of price retraction over the 102 postlaw price cycles.22 In \"All three price retractiom (onjurre 16, 2001, November 3, 2001, and. April 25, 2003) were by HP and were in full. The price retraction on November 3, 2001, by HP diifers from the other mo in that HP alone hiked its price just the day before. 1005 JOURNAL OF POLITICAL ECONOMY two cases, BP, as the price leader, frilly retracted its price hike on the third day after either Caltex or Shell failed to follow on the second day. Note that even in these two cases, BP kept its price hike on the second day. Over the rest of the price cycles, Caltex or Shell, if not a leader, always followed the leader on the second day. However, BP appears to receive special treatment: it did not follow on the second day in ve cases, but neither Caltex nor Shell retracted prices in response. Mobil mostly followed on the second day, and the independent rms largely followed on the third day. 3. Drastic Changes in Price Leadership Pattern As in gure 2, the price leaders for the postlaw cycles can also be clearly identied.\" Figure 6 displays the leaders of the 102 price cycles between May 10, 2001, and October 21, 2003. Caltex, never before a leader, now initiates price hikes for 52 of the 102 cycles. BP, almost always a leader before, now initiates price hikes for only 49 of the 102 cycles. There are seven mutually exclusive and exhaustive leadership types: (1) BP leads alone (27 cycles), (2) Caltex lads alone (37 cycles), (3) Shell leads alone {15 cycles), (4) BP and Caltex lead together (eight cycles), (5) BP and Shell lead together (eight cycles), (5) Caltex and Shell lead together (one cycle), and (7) BP, Caltex, and Shell all lead simulta- neously (six cycles). These observations are consistent with the hypoth- esis that price leadership under the law needs to be allocated among the rms. 4. Cycle Length Becomes More Unpredictable Under the law, the length of a price cycle becomes much more unpre- dictable. Dene the day on which the initiating price hike (s) took place as the stat": day of a price cycle and the day immediately before as the last day of the previous cycle. Table 1 shows the weekday frequency distribution of the cycle start day under the tinting law. Over a third of the postlaw cycles start on a weekday other than the three days on which the prelaw cycles always started. Dene the length of a cycle as the number of days from the start day through the last day of the cycle. Table 2 shows the postlaw cycle length 93 For 94 out of the 102 price cyclesI the price leaders are as clear-cut as those for the cycles shown in g. 2. The leaders for these 94 cycles are always BF, Caltex, or Shel]. None of the other rms in the market led any of these 94 cycles. For the other eight cycles, one or more independent rms had positive average price changes on the day when one or more of the three largest rms hiked price. Because the independent rms' price increases are much smaller in size, they are not considered as price leaders. There is a full price cycle between October 22, 2003, and the end of the sample period. This last cycle is ignored in the analysis because Mobil co-led this cycle. Shell Caltex BP . FIG. 6.-Price leadership pattern in the 102 wars of attrition under the law1008 JOURNJKL OF POLITICAL ECONOMY TABLE 1 DAY or WEEK FREQUENCY Drs'nuBUTroN or CYCLE START DAY UNDER THE Law DAY or? THE WEEK Monday Tuesday- chnesday Thursday Friday Saturday Sunday 1? Start day frequency T 25 20 19 5 1 25 102 distribution by leadership type. While the length of all prelawr cycles is between 5 and 9 days, the length distribution of the postlaw cycles is much more dispersed; about a third of the cycles have a length equal to or longer than 10 days. The length of postlaw cycles, even after breaking them down by leadership type, is still more dispersed than the length of all prelaw cycles. The average cycle length is not statistically dierent across three leadership types (BP leads alone, Shell leads alone, and multiple rms lead), and the average cycle length for the Caltex leading alone type is slightly bigger because the three longest cycles all happen to have been led by Caltex alone. While the null that the prelaw cycle length time series is generated by a white-noise process of tmcor- related random variables with a constant mean and a constant variance is soundly rejected by the Barlett periodogranrbased test or the Box and Pierce Q test for white noise (the 1tt-values are 0.0006 and 0.0000, respectively), the same null for cycle length under the law cannot be rejected by the same two tests (the pvalues are 0.15 and 0.84, respec- tively) . C. Impact .93\" Cost Changes Section [[I.B presented evidence that cost changes caimot explain the existence of the regular price cycles because the cycles continue to exist even when cost is monotonically decreasing and gasoline margins ex- hibit a cycle that is essentially identical to the retail price cycle. Sections IVA and [VB provide additional evidence. If cost changes are driving the price cycles, why should the regular price cycles disappear in the rst four months of the timing law? Furthermore, why should the price leadership pattern dier dramatically before and under the law? These facts are all consistent with the Edgeworth price cycle equilibrium. The rms face a war of attrition problem when price is in the competitive region, and this war of attrition game, which determines the timing of price hikes, is altered by the timing law. This explains the drastic change in price leadership patterns. Figure '1 in Section 111.3 suggests that the level of retail price varies with the level of import cost. Import cost a'ects the floor against which the retail price cycles bounce. The oor, in theory, is the marginal cost TABLE 2 FREQUENCY DISTRIBUTION OF CYCLE LENGTH UNDER THE LAW DAY 10 1-16 Cycles BP leads alone Cycles Caltex leads alone 37 AN Cycles Shell leads alone HONOT Cycles multiple firms lead All cycles 1021010 JOURNAL OF POLITICAL ECONOMY of supplying gasoline. Since variations in import cost track the changes in marginal cost of supplying gasoline, we can study how the variations in import cost a'ect the retail price cycle dynamics. The basic logic of Maskin and Tirole's (1938) model implies that cost changes can have subtle impacts on retail price. The core feature of the Edgeworth price cycle equilibrium is that price increases quickly and decreases gradually, and a war of attrition problem is embedded at the bottom of the price cycles. In their model, the falling phase of a cycle ends when the gasoline margin is zero or when a war of attrition starts. This implies that gasoline price and marginal cost at the end of a price cycle should be close. Thus, if marginal cost has decreased since the start of a price cycle, it would require a greater reduction of, and take a longer time for, retail price to be close to marginal cost again. This implies that cost changes between the start and end of a cycle aect the amplitude and length of this cycle. The amplitude of a price cycle can be measured by the height of either the rising phase or the falling phase. If marginal cost is constant over time, these two measures yield the same rault; if marginal cost changes over time, they yield different results. Dene the amplitude of the rising phase as the market average price at the top of the cycle minus the price at the end of the previous cycle, and dene the am- plitude of the falling phase as the price at the top of the cycle minus the price at the end of this cycle. Thus, the difference between the two cycle amplitude measuresthe height of the rising phase minus the height of the falling phaseis the same as the market price at the end of the current cycle minus the market price at the end of the previous cycle. The logic of the price cycle equilibrium thus implies the following testable implication: the di'erence between the two cycle amplitude measures varies positively with cost changes (the cost at the end of a cycle minus the cost at the end of the previous cycle). Figure 7 conrms this implication. The sample period in gure 7 is January 1 through October 31, 2003 (for which the import cost series is available). If the difference between the two measures of cycle am- plitude is regressed on a constant term and cost change, the coefcient on cost change is 0.83, with a standard error of 0.10. Cost change also a'ects the length of the price cycles (dened as the number of days between two lead price hikes) since it takes longer for the retail price to fall close to the marginal cost again if the marginal cost itself has decreased since the start of the cycle. This logic suggests another testable implication: the length of a cycle tends to be longer if the cost at the end of the cycle is lower than the cost at the start of the cycle. Figure 3 shows the relationship between cycle length and cost change over the postlaw period May 15, 2001, through October 21, 2003. The 5 I_ 0 -5 l Rising phase height - falling phm height d6 4 -2 D 2 4 Change In Import met FIG. \"itThe impact of cost changes on cycle amplitude, January l to October 31, 2003. The x axis is the import cost at the end of a cycle minus the import cost at the end of the previous cycle. The y axis is the height of the rising phase ofa cycle minus the height of the falling phase of the same cycle. This is the same as the market average price at the end of a cycle minus the market price at the end of the previous cycle. 15 l e 10 l -6 4 .2 u 2 4 Change in wholesale cost FIG. 8.The impact of cost changes on cycle length, May 15, 2001, to October 21, 2003. The x axis is the wholesale price paid by the BP retailer at the end of a cycle minus that at the beginning of the cycle. The length of a cycle, the y axis, is the number ofclays from the start through the end of a cycle. 1012 JOURNAL OF POLITICAL ECONOMY cost change is the wholesale price paid by the BP retailer at the end of a cycle minus the wholesale price paid at the beginning of the cycle. If cycle length is regressed on a constant and cost change, the cost change coefficient is -0.52, with a standard error of 0.12. If cost change is measured by the wholesale price paid by the small independent retailer, the cost change coefficient is -0.32, with a standard error of 0.11. If the sample period is restricted to January 1 through October 21, 2003, for which the import cost series is available, the coefficient for (import) cost change is negative (-0.25), though not statistically significant (p- value is 0.16). V. Mixed Strategies in Price Leadership The results in the previous section suggest that the Maskin-Tirole price cycle equilibrium captures the fact that a war of attrition game is at the bottom of each of the gasoline price cycles. In this section, I present evidence that the observed price leadership outcomes under the law are better characterized by mixed strategy plays than by two alternative hypotheses: (1) the firms play a pure strategy in each war of attrition or (2) they simply alternate as price leader over the wars. In Maskin and Tirole's (1988) model, firms always move sequentially and play the same mixed strategies in each war of attrition. This implies that there is always a single leader and the leadership outcome does not exhibit any serial correlation. The postwar price leadership patterns shown in figure 6 clearly reject these implications. Two or three firms may relent simultaneously, and a firm tends to be less likely to lead again if it led in the previous war. This is not surprising given that (1) the timing of the war of attrition under the law, as argued in Section II, is simultaneous instead of sequential, and (2) the gasoline firms have the incentive to coordinate over the wars of attrition. My point is that once these two factors are taken into consideration, the mixed strategy presumed by Maskin and Tirole can generate the postwar price lead- ership patterns. I hypothesize that the timing of the attrition wars is simultaneous and that the firms play the mixed strategy presumed in Maskin and Tirole's model (firm i always relents with probability p, on date t conditional on no firm having relented before then) within each war, but the probability with which each firm relents is affected by the outcome of the previous war. This hypothesis implies that the leadership outcome of each war, a random realization of mixed strategy play, must be random once con- ditional on the outcome of the previous war. In addition, the presumed mixed strategies impose strong restrictions on the leadership types and their frequencies. I consider these restrictions in subsection A and the(MIXED) STRATEGY IN OLIGOPOLY PRICING 1013 alternative hypotheses in subsection B. Here I present evidence that the outcome of this war, once conditional on the outcome of the previous war, is an independent draw from a random process. First note from gure 5 that the leadership outcome of this war, conditional on the outcome of the previous war, is far from determin- istic. For example, the leadership outcome of the 27 wars immediately preceded by a war in which BP led alone is that (1) BP leads alone twice, (2) Caltex leads alone 13 times, (3) Shell leads alone seven times, and (4) multiple rms lead simultaneously ve times. To test conditional serial independence, I separate the wars of attrition in the sample into four subsamples or four types according to the leadership outcome of the previous war: (1) BP alone led (27 wars), (2) Caltex alone led (37 wars), (3) Shell alone led (15 wars), and (4) multiple rms led together (22 wars). I use the nonparametric run test (see, e.g., Gibbons and Chakraborti 2003), which has been used in the recent literature that tests mixed strategies in sports games. The rim test of serial indepen- dence is based on the number of runs in a sequence?1 A small (large) number of runs indicates positive (negative) serial correlation. Let binary variable 11* equal one if event 3' is true for the lath war in a sample and zero otherwise. Event 3' indicates each of the seven leadership types or whether a rm (BP, Caltex, or Shell) is a leader. For example, If\TABLE S 1014 RUN TESTS OF SERIAL INDEPENDENCE WITHIN WARS OF ATTRITION OF THE SAME TYPE Number Number Number Number of 1's of Runs z-Statistic -Value of l's of Runs z Statistic -Value BP Leads Alone Previously: Caltex Leads Alone Previously: 27 Wars 37 Wars BP is a leader 1.54 -.19 Caltex is a leader -.53 .60 -1.86 06 Shell is a leader -.53 .60 1.45 .15 BP leads alone .63 -.64 52 Caltex leads alone .84 .40 .69 gregNinoON Shell leads alone -1.23 .26 80 BP and Caltex lead 59 56 BP and Shell lead F -.62 54 Caltex and Shell lead .28 BP, Caltex, and Shell lead 28 Shell Leads Alone Previously: Multiple Firms Lead Previously: 15 Wars 22 Wars BP is a leader -1.31 Caltex is a leader -.60 -1.73 08 Shell is a leader -2.55 -2.08 BP leads alone Caltex leads alone - 1.31 -.87 JOURNAL OF POLITICAL ECONOMY Shell leads alone -.94 BP and Caltex lead BP and Shell lead -2.55 Caltex and Shell lead BP, Caltex, and Shell lead NOTE.-The reported number of runs is the observed ones.(MIXED) STRATEGY IN OLIGOPOLY PRICING 1015 with at least one success if BP or Caltex led alone in the previous war. For the other two war types, the null cannot be rejected at the 5 percent level for 13 out of the 16 leadership sequences with at least one success, and none of the sequences are rejected at the 1 percent level. These results suggest that the leadership outcome of wars of attrition of the same type is serially independent. A. The Multinomial Distribution Test In this subsection, I derive and test the implications of the mixed strategy hypothesis on the leadership types and their frequencies. Note that only three firms (A, B, and C) potentially play the presumed mixed strategies since the three largest firms lead all the price cycles under the law. Qualitatively, the presumed mixed strategies imply that one of the fol- lowing seven mutually exclusive and exhaustive leadership types should arise in each war: (1) firm A leads alone, (2) firm B leads alone, (3) firm Cleads alone, (4) firms A and B lead together, (5) firms A and C lead together, (6) firms Band Clead together, and (7) firms A, B, and Call lead together. On a particular day of a war of attrition, it may be the case that none of the three firms relent. If such an event happens, the firms would continue to play the same mixed strategies until at least one firm relents. These seven price leadership types are precisely what we observe in Section IV.B.3. Quantitatively, the presumed mixed strategies imply specific multi- nomial distributions over the frequencies of the seven outcomes. Each war can be viewed as a random multinomial experiment with seven outcomes. Let firm i play relent with probability p, i = A, B, C. Then, the mixed strategy hypothesis implies that the multinomial distribution over the seven leadership types must be specified by three probability parameters (p., p, and p). For example, the probability with which firm A relents alone is pa(1 - pp)(1 - pc) + p.(1 - pa)(1 - pc) x [(1 -p.)(1 - pe)(1 - pc)] + ... + p(1 - pp)(1 - pc x [(1 - p)(1 - p)(1 - pa]'+ ... p.(1 - p=) (1 - pc) 1 - (1 - p)(1 -p=)(1 - pc)' (1) where [(1 - p.)(1 - p:)(1 - pc)] ' is the probability that none of the three firms has relented up to period t of a war. Similarly, the probabilities with which the other six leadership types arise are (1 - pa) pe(1 - pc) 1 - (1 - p)(1 - p;)(1 - pc)' (2)1016 JOURNAL OF POLITICAL ECONOMY (1 - p)(1 - pe)Pc (3) 1 - (1 -p)(1 -p=)(1 - pc) PA PB(1 - pc) 1 - (1 -p)(1 -p=)(1 - pc)' (4) pA(l - PB)Pc 1 - (1 -p)(1 - p=)(1 - pc)' (5) (1 - PA)DBPC 1 - (1 - p)(1 -p=)(1 - pc)' (6) and PAPEPC 1 - (1 -p)(1 -p=)(1 -pc) (7) While a typical multinomial distribution with seven outcomes has six free parameters, the multinomial distribution predicted by the pre- sumed mixed strategy equilibrium must be specified by three parameters PA> PB, and pc only. That is, the mixed strategy equilibrium imposes quite strong restrictions on the multinomial distribution. The Pearson x goodness-of-fit statistic can then be used to tes