Mixed Strategy in Oligopoly Pricing: Evidence from Gasoline Price
Cycles before and under a Timing Regulation∗
March 1, 2008
Zhongmin Wang
Department of Economics
Northeastern University
Boston, MA 02115
[email protected]
www.economics.neu.edu/zwang
Abstract
In their Edgeworth cycle equilibrium, Maskin and Tirole (1988) presume firms play mixed
strategies to decide price leadership at the bottom of each cycle. This paper takes this mixed
strategy assumption seriously. Nearly ideal price data shows the retail gasoline price in the Perth
area of Western Australia is well characterized by the price cycle equilibrium before and under a
unique simultaneous-move timing regulation. This paper then tests and finds evidence for the
hypothesis that the firms in this market play the presumed mixed strategies under the timing
regulation but not before. This result supports the idea that a player’s mixed strategy represents
other players’ uncertainty of that player’s pure choice.
Keywords: Mixed strategy, timing, Edgeworth cycle, war of attrition, price leadership
∗
I thank Kamran Dadkhah, Jim Dana, Juan Dubra, John Kwoka, Christian Rojas and seminar participants at Clark
University for helpful discussions and comments. I am especially grateful to an anonymous referee and the editor
whose comments on a previous paper inspired this new paper. Any errors are mine only.
1.
Introduction
In the Perth metropolitan area of Western Australia, firms used to be able to change retail
gasoline price at any time they wish, but a law that took effect in January 2001 forces them to set
price simultaneously (not knowing rivals’ price) and once every 24 hours.1 This paper begins by
documenting that the retail gasoline price in this market, both before and under the unique timing
regulation, is well characterized by the Edgeworth cycle equilibrium of Maskin and Tirole (1988).
At the bottom of each cycle in this equilibrium, firms face a war of attrition problem: which is first
to increase price? Maskin and Tirole presume firms play standard mixed strategies to resolve this
game. The purpose of this paper is to take this mixed strategy assumption seriously.
My hypothesis, which is supported by the empirical evidence, is that the firms in the Perth
market play the presumed mixed strategies under, but not before, the simultaneous-move timing
regulation. To motivate this hypothesis, consider the game of penalty kicks in soccer. A recent
literature finds professional athletes play mixed strategies in soccer penalty kicks and tennis serves
(Walker and Wooders 2001, Chiappori, Levitt and Groseclose 2002 and Palacios-Huerta 2003).
Athletes play mixed strategies in penalty kicks partly because of the simultaneous timing of the
game: a player has to act not knowing his rival’s action. Imagine the rule of the game is changed
so that the goalkeeper can observe and react to the kicker’s action. One would not expect mixed
strategies to be played anymore since the uncertainty caused by not knowing the rival’s action no
longer exists. Indeed, according to the Bayesian view of mixed strategy2 (Harsanyi 1973, Aumann
1987), a player’s mixed strategy represents other players’ uncertainty of that player’s pure choice.
More specifically, “players do not randomize; each player chooses some definite action. But other
1
This law, called the 24-hour-rule, is made technically possible by the internet. See section 3 for details of the law.
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 and 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 main rationales for mixed strategy.
2
1
players need not know which one, and the mixture represents their uncertainty, their conjecture
about his choice.” (Aumann and Brandenburger 1995, p. 1162). Under the timing regulation, the
firms in the Perth gasoline market are uncertain about rivals’ pure action when deciding whether to
hike their own price. Before the law, however, the uncertainty does not exist because the firms can
observe and react to rivals’ price when deciding their own.
The war of attrition game at the bottom of each Edgeworth cycle, like most economic and
social games, is a nonzero-sum game in which agents do not have the incentive to deliberately
randomize (Schelling 1960, p. 175).3 This is a major reason many find the use of mixed strategies
in such games counterintuitive and objectionable. The Bayesian justification is appealing, but
empirical evidence for or against equilibrium mixed strategies in nonzero-sum games remains
essentially nonexistent in the literature.4 One reason for the lack of evidence is simply that typical
real-life strategic settings are not suitable for conducting empirical tests of equilibrium mixed
strategies: timing is rarely imposed exogenously, strategy space is vast, and outcomes are
numerous. The timing regulation and the war of attrition game at the bottom of the gasoline price
cycles, however, provide a unique natural setting that overcomes these complications.
The strong regularities of the gasoline price cycles and the strategic force underlying them
are well captured by the Edgeworth price cycle equilibrium. The evidence comes from a nearly
ideal panel data set that tracks the price changes of nearly every gasoline site in the Perth area
before and under the law. To see the close resemblance between the gasoline price cycles and the
Edgeworth cycle, consider three figures. Figure 1 shows an example of the Edgeworth price cycle.
Figure 2 shows six price cycles before the law, using the hourly (from 6am to 6pm of each day)
brand average (regular unleaded) gasoline prices of three firms for a period of 39 days. Figure 3
shows seven cycles under the law, using the daily brand average retail prices of three firms and the
3
4
Players randomize in zero-sum games to conceal intention (von Neumann and Morgenstern 1944, p. 146).
A literature estimates game-theoretical models that may involve mixed strategies (e.g., Hendricks and Porter 1988).
2
daily wholesale price for a period of 57 days. A critical feature in all three figures is that firms
hike price sequentially, so a war of attrition problem is at the bottom of each and every cycle: price
increases by all would benefit all, but none would like to be the first to relent (i.e., hike price).
6
Figure 1: Maskin and Tirole (1988) Edgeworth Price Cycle
Firm 2 price
0
1
2
3
4
5
Firm 1 price
1
3
5
7
9
11
13
15
17
19
21
23
25
27
Time
Caltex
Mobil
90
95
BP
85
Price: Australian cents per liter
100
Figure 2: Hourly Brand Average Prices over Six Cycles before the Law,
07/04/00 4 pm to 08/12/00 9 am
0
100
200
300
400
500
Hour
85
80
Price: Australian cents per liter
90
Figure 3: Daily Brand Average Prices over Seven Cycles under the Law
BP
08/27/02
09/03/02
09/10/02
09/17/02
Caltex
Shell
09/24/02
Wholesale price
10/01/02
10/08/02
10/15/02
10/22/02
It is a simple matter to verify that the firms did not play mixed strategies to decide price
leadership before the law: one of the largest firms was nearly always the leader. The empirical task
is to show that under the timing law the firms play the standard stationary mixed strategies in wars
of attrition presumed by Maskin and Tirole. This task is possible because of the special properties
of the war of attrition game. First, this war of attrition game can be analyzed in isolation of the rest
of the price cycle. Second, the actions in this game are discrete (either relent or fight) and the
3
outcome is the identities of the price leaders. Third, the same war of attrition game is repeated
many times, providing rich variations in the observed outcome. Indeed, price leadership under the
law is distributed among the three largest firms and seven leadership types are observed: three
types in which an individual firm leads, three types in which two firms lead together, and one type
in which all three firms lead together. We can then use the variations in the leadership outcome to
test the mixed strategy hypothesis in the same spirit as the recent literature that tests mixed
strategies in sports games. The key departure is to account for the firms’ coordinating behavior
over the repeated wars of attrition. Players in nonzero-sum games have the incentive to tacitly
coordinate over repeated interaction while players in zero-sum games do not.
The empirical evidence verifies that the leadership outcome of each war, a random
realization of the presumed mixed strategies, is in fact random. The evidence indicates that the
stochastic regularities of the leadership outcome, including the frequency distribution and the
time series properties, are well captured by the presumed mixed strategy equilibrium. In
particular, the presumed equilibrium imposes strong restrictions on the frequency distribution of
the seven leadership types: while a typical multinomial distribution with seven outcomes has six
free parameters, the multinomial distribution over the seven leadership types is specified by
three parameters only (the probabilities with which the three firms each play relent).
This paper thus presents perhaps the first case study where agents have the incentive not to
randomize, but their plays in a strategic setting are well characterized by equilibrium mixed
strategies when they have to act without knowing rivals’ action. The finding provides some
empirical support for the Bayesian view of mixed strategy. This finding also suggests agents’
plays are unlikely to be characterized by equilibrium mixed strategies in strategic settings where
they can observe and react to rivals’ action. However, even in those settings, mixed strategy may
serve as a useful technical device, since the Edgeworth cycle equilibrium effectively captures the
4
gasoline price cycles before the law. Gasoline price cycles are not unique in the Perth market.
Castanias and Johnson (1993) note the gasoline price cycles in many U.S. cities in the 1960s
resemble the Edgeworth cycle. Recently, a few papers find the gasoline price cycles in many
Canadian cities resemble the Edgeworth cycle (e.g., Eckert and West 2004 and Noel 2007). Not
observing a timing regulation nor being free from data constraints, this literature does not address
the issue of whether firms play mixed strategies.
The rest of the paper is organized as follows. Section 2 discusses the key features of the
Edgeworth cycle equilibrium, the potential impact of the timing regulation, and the proposed
empirical tests of the presumed mixed strategies. Section 3 describes the timing regulation and
the data set. Sections 4 and 5, respectively, document the gasoline price cycles before and under
the timing restriction. Section 6 presents the test results. Section 7 concludes.
2.
Edgeworth Cycle, War of Attrition, and Mixed Strategy
Maskin and Tirole (1988) show the Edgeworth price cycle can emerge as a Markov
perfect Nash equilibrium in a market in which two firms compete repeatedly over the price of an
identical product. The intuition behind this equilibrium is that “if firms were stuck in the
competitive price region, …, a firm could raise its price dramatically and lure its rival to charge
a high price for at least some time….” (Tirole 1989, p. 256). Each price cycle, as those shown
in figure 1, has three distinct phases. If price is at marginal cost, the two firms are in the war of
attrition phase: both firms would like to increase price, but neither would like to be the first to
do so. The leader loses market share temporarily while the follower gets a free ride. The public
good of price leadership, however, must be provided in order for the cycle equilibrium to arise.
Maskin and Tirole presume firms use the mechanism of mixed strategy to allocate leadership.
This technical assumption is further elaborated below. The war of attrition game is identical
5
across all cycles. Once one firm relents by hiking price, the other firm reacts with a slightly
smaller increase in the following period. These two successive price hikes constitute the rising
phase of the cycle. In the subsequent falling phase, the two firms undercut each other gradually
until price reaches marginal cost. The rising and falling phases of all cycles are identical, and
firms do not play any mixed strategies in these two phases.
This cycle equilibrium arises in a stationary and deterministic market, and is not driven
by changes in marginal cost or market demand. Thus, with a nearly ideal price data set, it is a
simple exercise to examine whether this cycle equilibrium captures the key features of the
pricing dynamics observed in a market. In particular, if firms hike price sequentially, they face
the issue of which firm is first to hike price, thus a war of attrition results.
In the price cycle equilibrium, firms set price alternatingly so that a firm observes and
reacts to rivals’ price when deciding its own. This model setup fits the strategic setting in the
Perth gasoline market before the law. A firm can observe and react to rivals’ price before the
law because the firms are expected to charge the same price for at least a short period of time:
exogenous factors (such as menu cost) prevent firms from changing price every hour or minute.
As mentioned, Maskin and Tirole presume firms play the standard stationary mixed
strategies in wars of attrition to decide price leadership. Specifically, when the rival’s price is
set at the competitive level, a firm’s best response is to attach positive probabilities to both
relent (hiking price) and fight (keeping price at the competitive level). In any period of a war of
attrition, a firm is indifferent between relent and fight, with the expected value of both actions
given by the present discounted value of profits from that period on. Mixed strategy is a
convenient technical device here. By playing mixed strategies, both firms have a chance to be
the leader for any cycle so the burden of price leadership will be shared by the firms over time.
Nonetheless, in the model or in the Perth market before the law, firms observe and react to
6
rivals’ action when deciding their own, so the uncertainty of not knowing rivals’ action does not
exist. Therefore, the firms in the Perth market were unlikely to play mixed strategies before the
law. This hypothesis is reinforced by the fact that the firms have incentives not to play mixed
strategies: pure strategy ends a war immediately but mixed strategies may lead to costs of delay.
2.1
Potential Impact of the Timing Regulation
The timing regulation forces the firms in the Perth market to set price without knowing
rivals’ price. How can the firms continue to coordinate on the cycle equilibrium? Given the
firms coordinated on the cycle equilibrium before the law, it is quite natural for them to continue
to do so under the law. That is, the cycle equilibrium serves naturally as the focal equilibrium.
Indeed, the price data suggests firms started to initiate price hikes and move toward the cycle
equilibrium in about a month and half after the law took effect (see Appendix A). It appears
that not knowing rivals’ price is not a big obstacle to reproducing the rising and falling phases of
the cycle equilibrium. Suppose firm A hikes price to a high level on date t . When deciding
their price for date t + 1 , the rival firms do not observe firm A’s date t + 1 price, but they can
expect firm A to continue to charge a high price on date t + 1 . A price hike represents a credible
short-run commitment that arises endogenously: it is an action (not an announcement) that
incurs short-term losses in market share and its strategic purpose is defeated if firm A does not
continue to charge a high price. Similarly, when a firm decides price during the falling phase, it
can expect rivals will not rush back to the competitive level.
On the other hand, the timing of the war of attrition game under the law becomes truly
simultaneous: no credible short-run commitment exists when price reaches the competitive
level. Hence, firms are naturally uncertain about each other’s pure choice during this phase. A
firm decides to either relent or fight on a particular day, but rival firms do not observe the firm’s
7
specific action. Since the rationale for mixed strategy exists, it is natural to hypothesize that the
firms in the Perth market play the presumed mixed strategies under the timing regulation.
The firms still have the incentive to tacitly coordinate over the wars. The best the firms
can do is to coordinate a pure strategy equilibrium in each war so that at least one firm relents
immediately without any cost of delay. To do so, however, the firms must be certain about each
other’s pure choice at the beginning of each war, which is rather implausible under the law.
Firm-specific private information (about cost or demand or other aspects of firm operations that
affect payoff) always exists. What further complicates the firms’ ability to coordinate on pure
strategy equilibria is the fact that seven price leadership types are observed. If all seven
leadership types result from pure strategy equilibria, the firms face the difficult task of
coordinating on one of the seven pure strategy equilibria in each war.
The timing law does not prevent firms from coordinating, though. For example, it seems
logical that if firm A led alone in the previous war, rival firms may expect firm A to play relent
with a smaller probability in this war so that firm A is less likely to lead again. Coordinating in
this way does not require firms to be certain about rivals’ choice of pure strategy. They only
need to update their beliefs about the probability with which each firm plays relent in the current
war, after observing the leadership outcome of the previous war. Indeed, this paper finds that a
firm is less likely to be a leader again if it led alone in the previous war.
2.2
Empirical Tests of Mixed Strategies
The question of whether the firms play mixed strategies or pure strategies is ultimately
an empirical one. This subsection outlines the two main empirical tests of mixed strategies.
If the firms play mixed strategies, the first implication is that the leadership outcome of
each war must be random. That a firm is less likely to be a leader again if it led alone in the
previous war indicates the outcome of the previous war alters beliefs about the probability with
8
which each firm plays relent in the current war. The outcome of this war, still a random
realization of the players’ presumed mixed strategies, must still be random, conditional on the
outcome of the previous war. Since this property is confirmed, the wars of attrition under the
law can be separated into different types according to the outcome of the previous war, and wars
of attrition of the same type can be viewed as independent.
The second prediction tested is a unique property of the presumed stationary equilibrium
mixed strategies: firm i always relents with probability pi in period t conditional on no firm
having relented before then.5 Since the three largest firms lead all the price cycles under the law, it
is reasonable to assume that three firms ( A, B, C ) play the presumed mixed strategies. When the
timing is discrete and simultaneous, this mixed strategy equilibrium has strong predictions on the
leadership types and their associated frequencies.
The qualitative prediction is that one of the following 7 mutually exclusive and exhaustive
leadership types should arise in each war: (1) firm A leads alone; (2) firm B leads alone; and (3)
firm C leads alone; (4) firms A and B lead together; (5) firms A and C lead together; (6) firms B
and C lead together; and (7) firms A, B and C all lead together. Thus, each war can be viewed as a
random multinomial experiment with seven alternative outcomes.
Moreover, the mixed strategy equilibrium predicts a specific multinomial distribution over
the seven possible outcomes. For example, the probability with which firm A relents alone is:
p A (1 − pB )(1 − pC ) + p A (1 − pB )(1 − pC ) * [ (1 − p A )(1 − pB )(1 − pC ) ] + ...
(1)
+ p A (1 − pB )(1 − pC ) * [ (1 − p A )(1 − pB )(1 − pC ) ] + ...
t
= p A (1 − pB )(1 − pC ) / [1 − (1 − p A )(1 − pB )(1 − pC ) ]
where [(1 − p A )(1 − pB )(1 − pC )] t 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
5
Benoît and Dubra (2007) study this equilibrium in N-player wars of attrition.
9
(2)
(1 − p A ) pB (1 − pC ) [1 − (1 − p A )(1 − pB )(1 − pC ) ] ,
(3)
(1 − p A )(1 − pB ) pC [1 − (1 − p A )(1 − pB )(1 − pC ) ] ,
(4)
p A pB (1 − pC ) [1 − (1 − p A )(1 − pB )(1 − pC )] ,
(5)
p A (1 − pB ) pC [1 − (1 − p A )(1 − pB )(1 − pC ) ] ,
(6)
(1 − p A ) pB pC [1 − (1 − p A )(1 − pB )(1 − pC ) ] , and
(7)
p A pB pC [1 − (1 − p A )(1 − pB )(1 − pC ) ] .
While a typical multinomial distribution with seven outcomes has six free parameters, the
multinomial distribution predicted by the mixed strategy equilibrium is specified by three
parameters p A , pB , and pC only. That is, the mixed strategy equilibrium imposes restrictions on
the multinomial distribution. Given that wars of attrition of the same type can be viewed as
independent and identical, we can estimate the three parameters. The Pearson chi-square goodness
of fit statistic (with 3 degrees of freedom) can then be used to test whether the observed frequency
distribution is the multinomial distribution predicted by the mixed strategy equilibrium.
This multinomial distribution test is similar to the method used by the recent literature to
test the generic indifference property in zero-sum games. Walker and Wooders (2001), for
example, estimate the winning probabilities of two discrete actions (e.g., serving the tennis ball to
the right or left), which are taken as the parameters of two binomial distributions, and then use the
chi-square statistic (with 1 degree of freedom) to test whether the two binomial distributions are the
same. Unfortunately, it is not feasible to test the indifference property in a market setting because
the firms’ expected value of an action (relent or fight) is not observable. Because of players’
incentive to coordinate, the expected value of an action in any repeated nonzero-sum games is hard
to evaluate (because it should not be evaluated over a single constituent game).
10
In addition to the above two main tests, this paper also presents evidence that Markov
switching among the different types of wars of attrition captures the time series properties of the
observed price leadership patterns under the timing regulation.
3.
The Law and Price Data
The law, called the 24-hour-rule, requires that every gasoline site in the Perth area must (1)
notify the government of its next day's retail price by 2:00 pm today so that the notified price can
be published on an internet website (www.fuelwatch.wa.gov.au), and (2) post the published price
on its price board at the start of the next day for a period of 24 hours. The first requirement forces
the timing of price changes to be synchronous since the firms do not observe rivals’ price when
deciding their own. The second requirement forces the frequency of price changes to be at most
once every 24 hours. See the internet website for details of the 24-hour-rule.6
Because of the 24-hour-rule, I am able to download from the website an ideal panel data set
that has the daily price record of all retail sites in the market from the start of the rule January 3,
2001 through October 31, 2003.7 During this period, Caltex, BP, Shell and Mobil, the four oil
firms in the market, operate or control an average of 88, 67, 46, and 23 sites per day, respectively.
Gull and Peak, the two main independent brands, operate an average of 38 and 18 sites per day,
respectively. These six brands account for about 85% of the retail sites in the Perth gasoline
market. The daily market average price from the start of the 24-hour-rule through October 30,
2003 is shown in Appendix B.
6
The internet website mentions “motorists' frustration at intra-day price fluctuations and the significant difference
between city and country fuel prices” were the political reasons that led to the 24-hour-rule. There was a loophole to
the 24-hour-rule prior to August 24, 2001. During that period, a station must nominate its next-day retail price, but is
not required to move to the nominated price. A station could switch, during the day, between the nominated price and
the previous day’s retail price without violating the law. Since the major firms in the market did not take advantage of
this loophole, it does not affect the analysis in this paper.
7
The wholesale price used in figure 3 was provided to the author by a small independent gasoline firm in the Perth
market. It is the wholesale buying price paid by this firm to a major oil firm.
11
To study the price cycles before the law, I use a high frequency panel data set that covers
the retail price of nearly every gasoline site in the Perth area for the period July 1, 2000 to
December 20, 2000.8 For three brands (BP, Caltex and Mobil), the price data is hourly, between
the hours of 6 am and 6 pm, seven days a week. These hourly prices were electronically
sourced from purchase transactions with gasoline credit cards.9 For Shell and the independent
brands, retail prices were collected (via drive-by) twice a day, between 5 am and 9 am, and noon
and 3 pm, Monday to 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 pre-law data
set covers 286 stations: 89 Caltex sites, 73 BP, 35 Shell, 26 Mobil, 30 Gull, 7 Peak, and 26 sites
of several small independent brands.
4.
Price Cycles before the Law
This section presents evidence that the gasoline price cycles before the law are well
explained by the Edgeworth cycle equilibrium, but the firms did not play mixed strategies to
decide price leadership. There are 21 price cycles for the sample period before the law. As
shown in figure 2, the brand average prices increase quickly and decrease gradually. The firms
hike their prices sequentially, so they face a war of attrition problem at the bottom of each cycle.
These patterns are cleanly captured by the price cycle equilibrium. Below, I explore the stationlevel price data to offer additional discerning evidence.
The Edgeworth cycle equilibrium is generated by oligopoly strategic interaction, but
there are hundreds of gasoline sites in the Perth market. The key to reconcile this discrepancy is
to recognize the price setters in this market are the few oil and independent firms, not the
8
The data was collected by Informed Sources, a market research firm in Australia.
Many drivers in Australia purchase gasoline using gasoline credit cards (e.g., BP Plus Card and Caltex Star Card).
Each time a gasoline credit card is used to purchase gasoline at a retail site, the retail price at the pump was sent
electronically to Informed Sources. The price data provided to the author is the latest price for each station at each
hour between 6 am and 6 pm.
9
12
individual gasoline sites. The oil and independent firms synchronize and homogenize intrabrand
price increases. Without intrabrand synchronization, we would not observe the clear sequential
patterns of price hikes in figure 2. A firm homogenizes intrabrand price increases by hiking the
prices at most of its gasoline sites to a single price. To see directly intrabrand synchronization
and uniformity, consider as an example the rising phase of the first price cycle shown in figure
2, which took place on July 13, 2000. The average BP price increased from 87.2 cents per liter
at 11 am to 88.9 at 12 noon, and then to 91.9 at 1 pm. The average increase from 87.2 to 88.9
took place because the price of 13 BP sites was hiked from 86.5 to 92.9 between 11am and noon
while the other 60 BP sites’ prices were not changed. The average increase to 91.9 occurred
because another 35 BP sites’ prices were hiked to 92.9 (mostly from 86.5) between noon and 1
pm. Thus, between 11 am and 1 pm, 48 of the 73 BP sites hiked price to 92.9 (mostly from
86.5). Because not a single site of the other brands had hiked price by 1 pm that day, we
observe strong, albeit not perfect, intrabrand synchronization and uniformity in price hikes.10
The leader for each of the six cycles shown in figure 2 (or the other cycles in the pre-law
sample) is clearly identified. BP was the first to hike price in 18 of the 21 cycles, and Shell was
the leader for the other three cycles. In the three cases where Shell was first to hike price, BP
started to hike price within an hour. Caltex, the largest firm in the market, was never the first to
hike price before the law. Since Caltex served as a leader most often under the law, I do not
consider this pre-law price leadership pattern to be consistent with mixed strategy play. Rather,
this leadership pattern probably reflects BP’s position as the market leader. BP’s loss in market
share before the law is rather small because major rivals typically responded within a few hours,
and BP retracted its price hike if major rivals did not follow quickly enough.
10
Price decreases typically do not exhibit strong, if any, intrabrand synchronization or uniformity.
13
Price retractions, visible in figure 2, result directly from wars of attrition. Indeed, price
retractions are part of the Edgeworth cycle equilibrium in a 3-player model (Noel 2006).11
Gasoline price retractions may be partial or full. For example, on July 13, 2000, the average BP
price decreased by 1.3 cents between 4 pm and 5 pm because 14 of the BP sites that had hiked
price to 92.9 retracted their price hikes and returned to the pre-hike price levels. The temporal
price retractions by BP over six other cycles are much more pronounced because more BP sites
retracted price hikes, and in two of these six cases, the retraction is full in that all sites returned to
the pre-hike levels. Why are some retractions partial and others full? A firm is happy to stop
retracting if rival firms started to follow quickly and the retraction is full if rivals did not follow.
The following details about the pre-law price cycles are informative when compared
with the price cycles under the law. The leader of a price cycle always started to hike price
between 11 am and 2 pm on Tuesday (5 cycles), Wednesday (8 cycles) or Thursday (8 cycles).
The length of a price cycle, defined to be the period between two lead price hikes, is six days (8
cycles), seven days (7 cycles), eight days (3 cycles) or nine days (3 cycles). Caltex typically
followed the leader within two or three hours, and Shell typically followed BP within a few
hours. BP retracted its price hike significantly or fully over six cycles, and in these cases, either
Caltex or Shell did not follow quickly. The hour at which Mobil, Gull or Peak followed the
leader tends to be less precise, but over the vast majority of the 21 cycles, Mobil followed by 6
am of the second day, and Gull and Peak followed by 9 am of the second day.12 The oil firms
11
In such a model, after one firm relents, the two remaining firms may not follow immediately because they still have
the incentive to be the last to hike price. Therefore, the first firm may retract its hike.
12
For about half of the cycles Mobil followed the leader within a few hours, but for the other cycles, the data is such
that Mobil started to hike price at 6am the next day. Because the data between 6pm and 6am are not available, it is
possible that Mobil may have started to hike price before 6am. In the data, Gull and Peak always started to hike price
on the second day, mostly between 7am and 9am. 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 indicates.
14
tend to hike price to match the price leader while the independent firms tend to slightly undercut
the price leader (typically by 0.2 cents).13
5.
Price Cycles under the Timing Regulation
Regular gasoline price cycles disappeared after the timing restriction took effect on
January 3, 2001, but reappeared in early May 2001. There are 103 regular gasoline price cycles
from May 10, 2001 through October 21, 2003. These price cycles under the law, as those
shown in figure 3, are also well characterized by the Edgeworth cycle.
Since the firms still hike price sequentially, they face the war of attrition problem at the
bottom of each cycle. In fact, the disincentive to be a leader is much bigger under the law: a
leader has to lose market share to its rival firms for at least 24 hours before the rival firms can
respond. This implies that the public good of price leadership needs to be distributed among the
firms. The brand average prices are now typically hiked in a single step. This is due to stronger
intrabrand synchronization in price hikes, which, in turn, is due to the need to increase price
quickly. A two-step price hike takes only a few hours before the law but would take 48 hours to
implement under the law. Price hikes are very rarely retracted under the law, which is not
surprising since temporary retraction that lasts a few hours cannot exist under the law. There
are only two cases of price retraction over the 103 price cycles under the law. In both cases BP
was the price leader and BP fully 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. As argued before, if a leader is not committed to keeping its price hike
on the second day, the strategic purpose of hiking price would be defeated. Over the rest of the
13
For example, on July 13, 2000, after BP hiked most of its sites’ price to 92.9, Caltex, Shell and Mobil all hiked
most of their sites’ price to 92.9, but Gull and Peak hiked most of their sites’ price to 92.7. Hence, the independent
firms are the first to cut price.
15
price cycles, Caltex and Shell, if not a price leader, always followed the leader on the second
day. BP, if not a leader, almost always followed the leader on the second day. Mobil followed
mostly on the second day, and the independents followed largely on the third day. Hence, the
order of price followership under the law is similar to that before the law.
It is useful to take a brief look at the period of adjustment that followed the enactment of
the law. Appendix A shows the brand average prices of Caltex, BP and Shell from January 3
through May 21, 2001. Shell and Caltex started to initiate price hikes in the middle of February.
Even during this period, the leader maintained its price hike on the second day. During this
adjustment period, however, the leader almost always retracted its price hike on the third day
because one of the three largest firms did not fully follow on the second day. When the lead
price hike was matched by the major rivals, the regular price cycle equilibrium re-emerged.
5.1
Drastic Changes in Price Leadership Pattern
As shown in figure 3, the price leaders for the cycles under the law can be clearly
identified. Figure 4 displays the price leaders of the 102 price cycles between May 10, 2001 and
October 21, 2003.14 Caltex, never a leader before, is now a leader for 52 of the 102 cycles. BP,
almost always a leader before, is now a leader for only 49 of the 102 cycles. Shell, leader of 3
of the 21 cycles before the law, is now a price leader for 30 cycles. There are seven mutually
exclusive and exhaustive leadership types: (1) BP leads alone (for 27 cycles), (2) Caltex leads
alone (37 cycles), (3) Shell leads alone (15 cycles), (4) BP and Caltex lead together (8 cycles),
14
For 94 out of the 102 price cycles, the price leaders are as clear cut as those for the cycles shown in figure 3. The
leaders for these 94 cycles are always BP, Caltex or Shell. None of the other firms in the market led any of these 94
cycles. For the other 8 cycles, one or more independent firms had positive average price changes on the day when one
or more of the three largest firms hiked price. Because the independent firms’ price increases are much smaller in size,
they are not considered as price leaders. For example, Caltex raised its average price by 6.95 cents on February 19,
2003 to start a new cycle, and Peak raised its average price by 1.56 cents on the same day. Peak is not viewed as a
leader for this cycle, especially after considering that Peak followed Caltex with an increase of 5.01 cents on February
21, 2003. 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.
16
(5) BP and Shell lead together (8 cycles), (6) Caltex and Shell lead together (1 cycle), and (7)
BP, Caltex and Shell all lead simultaneously (6 cycles). These observations are consistent with
the qualitative implication of the standard stationary mixed strategy equilibrium in wars of
attrition with discrete and simultaneous moves.
Shell
Caltex
BP
Figure 4: Price Leadership Pattern in the 102 Wars of Attrition under the Law
bp
1
3
2
5
4
7
6
9
8
caltex
shell
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102
War Number
Table 1: Day of Week Frequency Distribution of Cycle Start Day under the Law
Day of week
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Start day freq.
7
25
20
19
5
1
25
Days
Cycle length
5.2
1
Table 2: Frequency Distribution of Cycle Length under the Law
2
3
4
5
6
7
8
9 10 11 12 13 14
1
3
7 24 19 14 16 8
4
3
1
15
1
1-7
102
16
1
1-16
102
Cycle Length Becomes White Noise
Under the 24-hour-rule, the length of a price cycle becomes unpredictable, indicating the
firms are uncertain about when rivals would relent. Define the day on which the leaders of a
cycle hike price as the start or first day of a price cycle, and the day immediately before as the
last day of the previous cycle, and define the length of a price cycle as the number of days from
the start day through the last day. Over a third of the price cycles under the law start on a
weekday other than Tuesday, Wednesday or Thursday, the three days on which the price cycles
before the law always started. Table 1 shows the day of week frequency distribution of the
17
cycle start day under the law. While the length of all cycles before the law is between 6 and 9
days, the length distribution of the cycles under the law, which is shown in table 2, is much
more disperse. About a third of the cycles have a length equal to or longer than 10 days. The
null that the cycle length time series under the law is generated by a white-noise process of
uncorrelated random variables with a constant mean and a constant variance cannot be rejected
by the Barlett periodogram-based test or the Box and Pierce Q test for white noises (the p-values
are 0.16 and 0.84, respectively). However, the same null hypothesis for the cycle length before
the law is soundly rejected by the same two tests (the p-values are 0.0006 and 0.0000).
6.
Test Results
The evidence in the previous two sections suggests that the regularities of the gasoline
price cycles before and under the 24-hour-rule are captured by the Edgeworth cycle equilibrium.
This section presents evidence that the observed price leadership patterns under the law are
consistent with the predictions of the presumed mixed strategy equilibrium in wars of attrition.
Section 6.1 verifies that the leadership outcome of a war, conditional on the outcome of
the previous war, is random. The central empirical issue then becomes whether the stochastic
regularities in the leadership outcome are captured by the presumed mixed strategies. Sections
6.2 and 6.3 provide evidence that the observed frequency and time series properties of the price
leadership outcomes are consistent with the predictions of the presumed mixed strategy
equilibrium. Section 6.4 argues that the evidence is inconsistent with pure strategy equilibria.
6.1
Conditional Serial Independence
Since the firms have the incentive to tacitly coordinate, the wars of attrition in the
sample are not expected to be serially independent. This subsection first shows that a firm is
less likely to lead again if it led alone in the previous war. This subsection then confirms that
18
the outcome of this war, once conditional on the outcome of the previous war, is an independent
draw from a random process.
To see how the leadership outcome of the previous war affects the outcome of this war,
consider the following linear probability model. Let the dependent binary variable lik equal 1 if
event i is true for the k th war and 0 otherwise. Event i indicates each of the 7 leadership types or
k
whether a firm (BP, Caltex or Shell) is a leader. For example, lBP
equals 1 if BP is a leader of the
k
k th war, and lBPalone
equals 1 if BP alone is the leader of the k th war. The independent variables are
k −1
k −1
k −1
, lCaltex
the leadership types of the previous war ( lBPalone
alone ,..., l BP − Caltex − Shell ).
Since the independent variables in these regressions (without a constant term) are mutually
exclusive and exhaustive categories, the linear model is completely general: the estimated
coefficients are the probabilities that lik = 1 conditional on the leadership type of the previous war,
which are necessarily between 0 and 1. For example, BP leads 6 of the 27 wars that are preceded
immediately by another war in which BP relented alone, implying a conditional probability of
0.222 (=6/27), the first coefficient in regression (1) reported in table 3. Because the leadership type
of Caltex and Shell leading together has only one observation, it is combined with the leadership
type of BP, Caltex and Shell all leading together in regressions (1) through (3) in table 3. All four
multi-firm leadership types are grouped together in the remaining regressions reported in table 3.
This is because the Wald tests fail to reject the null hypothesis that the conditional probability with
which firm i is a leader is equal across the multi-firm leadership types in the previous war.
The estimated results from regressions (1) and (2) suggest that BP or Caltex is less likely to
lead again if it led alone in the previous war. For example, the probability that BP is a leader in
this war is only 22.2% if it led alone in the previous war, but the probability increases to 81.1% if
Caltex led alone in the previous war. Similarly, regressions (4) and (5) suggest that BP or Caltex is
19
less likely to lead alone again if it led alone in the previous war.15 The estimated coefficients in
regressions (3) and (6) suggest a similar pattern for Shell, but the pattern is not statistically
significant.
Table 3: Regression Analyses of Serial Dependence between Wars of Attrition
Independent variable
Frequency
with which
independent
variable
is true
Does
BP
lead?
(1)
Does
Caltex
lead?
(2)
Does
Shell
lead?
(3)
Does
BP
lead
alone?
(4)
Does
Caltex
lead
alone?
(5)
Does
Shell
lead
alone?
(6)
Do
multiple
firms
lead?
(7)
BP leads alone
in the previous war
27
0.222
(0.082)
0.556
(0.099)
0.444
(0.099)
0.074
(0.051)
0.481
(0.098)
0.259
(0.086)
0.185
(0.077)
Caltex leads alone
in the previous war
37
0.811
(0.066)
0.189
(0.066)
0.270
(0.075)
0.595
(0.082)
0.054
(0.038)
0.135
(0.057)
0.216
(0.070)
Shell leads alone
in the previous war
15
0.267
(0.118)
0.866
(0.091)
0.066
(0.066)
0.067
(0.066)
0.733
(0.117)
Multiple firms lead
in the previous war
22
0.091
(0.063)
0.500
(0.109)
0.091
(0.063)
0.318
(0.101)
BP and Caltex lead
in the previous war
8
0.250
(0.158)
0.625
(0.176)
0.250
(0.158)
BP and Shell lead
in the previous war
7
0.286
(0.176)
0.857
(0.136)
0.286
(0.176)
7
0.714
(0.176)
0.857
(0.136)
0.286
(0.176)
0.64
0.66
0.33
0.50
0.54
0.19
0.24
0.00
0.06
0.00
0.00
0.29
0.51
Either Caltex and Shell
lead together or BP,
Caltex, Shell all lead
Total frequency
R-squared
0.200
(0.106)
101
Wald tests
Null hypothesis
All coefficients equal
p-value
0.00
Coefficients for three
multi-firm leadership
0.11
0.51
0.98
categories are the same
Notes: Heteroskedastic-robust standard errors are in parentheses. Column 2 shows the number of wars for which
the corresponding independent variable is true. If Shell leads alone in the previous war, Shell does not lead alone
again in this war, thus the blank term in regression (6).
15
The estimated conditional probability with which firm i relents alone is smaller than the corresponding probability
with which firm i relents. The difference, of course, is the conditional probability with which firm i relents together
with at least one of the other two firms. Note that the coefficients in regressions (4) through (7), by row, sum to 1.
This is because one of the dependent variables in these four regressions must be true.
20
The size of the lead price hikes behaves very differently from the identity of the lead
price hikes. Since the size of the lead price hikes in the model is identical across all cycles, one
would expect the size of the lead price hikes observed in a market with random disturbances to
be generated by a white noise process. Indeed, the null that the time series of the lead price
hikes are generated by a white noise process cannot be rejected by the Barlett periodogrambased test or the Box and Pierce Q test for white noise.16 This finding supports the proposition
that the identity decision is independent of the size decision.
The estimated conditional probabilities in regressions (4) through (7) suggest the leadership
outcome of this war, conditional on the outcome of the previous war, is far from deterministic. For
example, the leadership outcome of the 27 wars immediately preceded by a war in which BP led
alone is (1) with probability 7.4% BP leads alone, (2) with probability 48.1% Caltex leads alone,
(3) with probability 25.9% Shell leads alone, or (4) with probability 18.5% multiple firms lead
together. The rest of this subsection tests whether the leadership outcome of this war, conditional
on the leadership type of the previous war, is an independent draw from a random process. For this
purpose, the 102 wars of attrition are separated into four subsamples or four types according to the
leadership type of the previous war: (1) BP alone led; (2) Caltex alone led; (3) Shell alone led; and
(4) multiple firms led together. The results in table 3 suggest the four multi-firm leadership types
can be grouped together when predicting the leadership type of the next war.
To test the null of conditional serial independence, I use the non-parametric run test (see,
e.g., Gibbons and Chkraborti 2003), which has been used in the recent literature that tests mixed
strategies in sports games. The run test of serial independence is based on the number of runs in a
16
The p-values in both cases are larger than 0.70. To conduct the tests, define a time series sequence to be the size of
the lead price hikes of the 102 wars. For the 23 wars in which multiple firms relent together, this sequence takes the
average of the multiple lead price hikes. The test results do not change if the sequence takes the maximum or the
minimum of the multiple lead price hikes. There are 131 relenting price hikes over the 102 price cycles, ranging from
2.57 to 13.43 cents per liter. The average size of a relenting hike is 7.85 cents per liter, and only four hikes have a size
smaller than 4 cents.
21
sequence.17 A small (large) number of runs indicate positive (negative) serial correlation. To gain
intuition about this test, consider the Bernoulli leadership sequences li = {li1 , li2 ,..., li102 } for the
entire sample of 102 wars. The regression results in table 3 indicate that BP or Caltex is less likely
to lead again if it led alone in the previous war, suggesting the sequences of lBP , lCaltex , lBPalone or
lCaltex alone are negatively serially correlated. Thus, one would expect a large number of runs in these
sequences and the run test would reject the null of serial independence for these sequences. On the
other hand, the regression results indicate this pattern does not hold for Shell, thus one would not
expect the run test to reject the null of serial independence for the sequences of lShell and lShellalone .
Let ri be the number of runs in the Bernoulli sequence li = {li1 , li2 ,..., liK } , where K is the
number of wars in a sample, and let Li = ∑ kK=1 lik be the number of successes in this sequence. The
expected number of runs in the sequence under the null of serial independence is
(8)
μr = 2 Li ( K − Li ) / K + 1 ,
and the variance is σ r2 = 2 Li ( K − Li ) [ 2 Li ( K − Li ) − K ] ⎡⎣ K 2 ( K − 1) ⎤⎦ . The test statistic,
z = (r − μr ) σ r , is approximately distributed as the standard normal distribution. As expected,
when the run test is applied to the leadership sequences li = {li1 , li2 ,..., li102 } of the entire sample, the
null of serial independence is rejected at normal significance levels for the sequences of lBP , lCaltex ,
lBPalone , and lCaltexalone , but not for lShell and lShellalone .
Table 4 reports the test results for the sequences of the four subsamples. The null of serial
independence cannot be rejected at the 5% level for any of the 18 leadership sequences with at least
one success if BP or Caltex led alone in the previous war (the p-values in all but one case are
bigger than 0.10). For the other two war types, the null of serial independence cannot be rejected
17
A run is a sequence of identical symbols. For example, the sequence (1, 1, 0, 1) has three runs.
22
at the 5% level for 13 out of the 16 leadership sequences with at least one success, and none of the
sequences are rejected at the 1% level. These results indicate the leadership outcome of wars of
attrition of the same type is serially independent.
Table 4: Run Tests of Serial Independence within Wars of Attrition of the Same Type
BP is a leader
Caltex is a leader
Shell is a leader
BP leads alone
Caltex leads alone
Shell leads alone
BP and Caltex lead
BP and Shell lead
Caltex and Shell lead
BP, Caltex and Shell
BP is a leader
Caltex is a leader
Shell is a leader
BP leads alone previously: 27
# of 1s
# of runs
z-stat p-value
6
13
1.54
0.12
15
13
-0.53
0.60
12
13
-0.53
0.60
Caltex leads alone previously : 37
# of 1s # of runs
z-stat p-value
30
12
-0.19
0.85
7
9
-1.86
0.06
10
19
1.45
0.15
2
5
0.48
0.63
13
15
0.20
0.84
7
9
-1.23
0.22
0
1
3
7
0.71
0.48
1
3
0.28
0.78
1
3
0.28
0.78
Shell leads alone previously: 15
# of 1s
# of runs
z-stat
p-value
4
5
-1.31
0.19
13
4
-0.60
0.55
1
2
-2.55
0.01
22
17
-0.64
0.52
2
5
0.40
0.69
5
10
0.26
0.80
3
7
0.59
0.56
3
6
-0.62
0.54
0
1
2
5
0.40
0.69
Multiple firms lead previously: 22
# of 1s
# of runs
z-stat p-value
9
13
0.62
0.54
17
6
-1.73
0.08
6
6
-2.08
0.04
BP leads alone
1
3
0.39
0.69
2
5
0.54
0.59
Caltex leads alone
11
5
-1.31
0.19
11
10
-0.87
0.38
Shell leads alone
0
1
2
4
-0.94
0.35
BP and Caltex lead
2
4
0.60
0.55
3
7
0.81
0.42
BP and Shell lead
1
2
-2.55
0.01
1
3
0.32
0.75
Caltex and Shell lead
0
1
0
1
BP, Caltex and Shell
0
1
3
7
0.81
0.42
Note: The reported number of runs is the observed ones. The expected number of runs can be computed by using
equation (8).
6.2
The Leadership Multinomial Distribution Test
This subsection tests whether the observed multinomial distribution over the seven price
leadership types is the same as the distribution predicted by the presumed mixed strategy
equilibrium. To emphasize, this distribution test is based on the idea that the presumed mixed
strategy equilibrium imposes restrictions on the multinomial distribution over the seven leadership
23
outcomes. If the presumed mixed strategies are played, the multinomial distribution must be
specified by three parameters, the probabilities with which the three firms each play relent on a day
of war of attrition. The predicted multinomial distribution is given by equations (1) through (7).
The findings in the previous subsection suggest wars of attrition of the same type may be
characterized by the same mixed strategy equilibrium.
The three parameters of the distribution can be estimated by the maximum likelihood
method. Suppose in a sample of wars of attrition of the same type, the observed frequencies of the
seven leadership types are ni , i = 1,...7 . Let n = ∑ i7=1 ni . Then, the likelihood function for the
multinomial distribution is
A( p A, pB , pC ; n, ni ) =
(
n!
n1 !n2 !...n7 !
) ⎡⎣
n
p A (1− pB )(1− pC ) ⎤ n1 ⎡
p A pB pC
⎤7
...
1−(1− p A )(1− pB )(1− pC ) ⎦
⎣ 1−(1− pA )(1− pB )(1− pC ) ⎦
The three parameters ( p A , pB , and pC ) can be obtained by solving the three first-order conditions:
(9)
n1 + n4 + n5 + n7
pA
−
n2 + n3 + n6
1− p A
n (1− pB )(1− pC )
A )(1− pB )(1− pC )
− 1−(1− p
(10)
n2 + n4 + n6 + n7
pB
−
n1 + n3 + n5
1− pB
− 1−(1− p
(11)
n3 + n5 + n6 + n7
pC
−
n1 + n2 + n4
1− pC
− 1−(1− p
=0
n (1− p A )(1− pC )
A )(1− pB )(1− pC )
=0
n (1− p A )(1− pB )
A )(1− pB )(1− pC )
=0
By substituting the estimated parameters ( pˆ A , pˆ B , and pˆ C ) into equations (1) to (7), we can obtain
the predicted frequencies of the seven price leadership types ( nˆi , i = 1,...7 ). The Pearson chisquare goodness of fit statistic is ∑ i7=1 (ni − nˆi ) 2 nˆi . We observe seven alternative outcomes, but
estimate only three parameters, thus the test has 3 degrees of freedom. To better understand the
intuition behind the test, consider the following proposition.
It can be easily shown the maximum likelihood estimates of the three parameters are
identical to the solutions to the following three equations:
24
(12)
pA / [1− (1− pA )(1− pB )(1− pC )] = (n1 + n4 + n5 + n7 ) n
(13)
pB / [1 − (1 − p A )(1 − pB )(1 − pC ) ] = (n2 + n4 + n6 + n7 ) n
(14)
pC / [1 − (1 − p A )(1 − pB )(1 − pC ) ] = (n3 + n5 + n6 + n7 ) n
The left hand side of equation (12) is the probability predicted by the mixed strategy equilibrium
that firm A is a price leader (alone or with other firms), and the right hand side is the observed
probability that firm A is a price leader. Equations (13) and (14) can be similarly interpreted.
Therefore, the information used to estimate the three parameters is four frequencies: the total
number of wars and the frequencies with which firm A, B and C each is a leader. The price
leadership multinomial distribution is characterized by seven frequencies. This is the reason why
the chi-square goodness of fit test has three degrees of freedom. Another useful observation from
equations (12) to (14) is that the probability with which firm i plays relent is proportional to the
frequency with which it is a leader. A firm is less likely to be a leader in a war because it relents
with a smaller probability in this war.
Table 5 reports by war type the estimated probabilities with which BP, Caltex and Shell
each play relent. Table 6 reports by war type the expected and observed frequencies of each
leadership type and the chi-square distribution test results. The null for two types of wars (Caltex
led alone or multiple firms led in the previous war) cannot be rejected at the conventional levels
(the critical values of the chi-square distribution with 3 degrees of freedom are 6.25 and 7.82 at the
10% and 5% significance levels, respectively). The null for the other two types of wars is rejected
at the 5% level, but not at the 2.5% level or higher (the critical value at the 2.5% level is 9.35).
The distribution test is stringent in the sense that slight changes in the observed frequencies could
change the chi-square statistic substantially. For example, the war type for which BP led alone in
the previous war has a chi-square statistic of 8.9. If the observed frequency of BP and Shell
25
together is reduced by 1 and correspondingly the frequencies of BP or Shell each leading alone are
increased by 1, the predicted distribution would not change (since the right hand sides of equations
(12) to (14) do not change at all), but the chi-square statistic would be reduced to 5.3.
Table 5: Estimated Probabilities with which a Firm Plays Relent by War Type
Prob. BP plays
relent
0.1118
Prob. Caltex
plays relent
0.2796
Prob. Shell
plays relent
0.2237
Caltex alone leads previously
0.5543
0.1293
0.1848
Shell alone leads previously
0.1800
0.5850
0.0450
Multiple firms lead previously
0.3264
0.6166
0.2176
BP alone leads previously
Table 6: Frequency Distribution of Leadership Types by War Type
In the current war,
BP leads alone
Caltex leads alone
Shell leads alone
BP and Caltex lead
BP and Shell lead
Caltex and Shell lead
BP, Caltex and Shell
Total frequency:
Chi-square statistic:
BP leads alone
Predicted Observed
3.4
2
10.3
13
7.7
7
1.3
0
1.0
3
3.0
1
0.4
1
27
27
8.9
In the previous war,
Caltex leads alone
Shell leads alone
Predicted Observed Predicted Observed
21.3
22
1.6
1
2.5
2
10.2
11
3.9
5
0.3
0
3.2
3
2.2
2
4.8
3
0.1
1
0.6
0
0.5
0
0.7
2
0.1
0
37
37
15
15
4.1
9.3
Multiple firms lead
Predicted Observed
2.7
2
9.0
11
1.5
2
4.3
3
0.8
1
2.5
0
1.2
3
22
22
6.4
The timing regulation and the special properties of the war of attrition game embedded in
the price cycle equilibrium free us from many of the complications that make typical oligopoly
markets poor settings for testing the mixed strategy hypothesis. However, decisions by a gasoline
firm in the Perth market are still arguably more complicated than an athlete’s decision in soccer
penalty kicks as many exogenous market factors (such as changes in crude oil price) potentially
affect firms’ relenting decisions. Therefore, the leadership distribution test is a joint test of the
presumed equilibrium mixed strategies and other implicit assumptions (such as changes in cost and
demand do not alter the firms’ mixing behavior significantly).
26
6.3
Markov Switching between Wars of Attrition
The results so far suggest the presumed mixed strategy equilibrium effectively captures
the leadership patterns observed under each of the four types of wars. This subsection presents
evidence that Markov switching among the four types of wars captures the time series (and
cross sectional) properties of the Bernoulli leadership sequences li = {li1 , li2 ,..., li102 } over the
entire sample of wars. That is, I hypothesize the observed price leadership data is generated by
the following stochastic process. Suppose the k th war is one of the four types. The outcome of
this war, which is a random realization of the estimated stationary mixed strategy equilibrium
for this type of war, determines the type of the (k + 1)th war. The outcome of the (k + 1)th in turn
determines the type of the next war and so on. This stochastic Markov switching process
incorporates the ideas that (1) the firms play the estimated stationary mixed strategies in each
war of attrition and (2) the probabilities with which each firm plays relent varies with the
outcome of the previous war because of tacit coordination.
The Markov restriction, motivated by the finding of conditional serial independence, is
the only time series information from the price leadership data that has been used to form the
stochastic switching process. Three properties of the leadership sequences are considered in
testing the hypothesis: the number of successes, the number of runs in each of the leadership
sequences li = {li1 , li2 ,..., li102 } , and the cross correlation coefficients among the three sequences of
lBP , lCaltex , and lShell . The number of runs in a sequence is determined by the number of
successes in the sequence and the degree of serial dependence in the sequence. The results in
section 6.1 suggest that the leadership sequences within each war type are serially independent,
but some of the leadership sequences over the entire sample of 102 wars are serially dependent.
27
To obtain the predicted empirical distribution of the three types of summary statistics, I
simulate the stochastic Markov switching process. Suppose the first war is of the type in which BP
led alone in the previous war (the results are not sensitive to the starting value). Then simulate the
outcome of the first war, given the estimated probabilities with which the three firms each play
relent in this type of war. The outcome of the first war then determines the type of the second war,
and so on. Continue this process until the 102nd war so that a single realization of the leadership
sequences is generated. A single point estimate of the summary statistics can then be obtained.
Then repeat the process 5,000 times to obtain the empirical distribution of the summary statistics.
Table 7 shows the predicted and observed number of successes for each of the leadership
sequences. The number of successes in each of the leadership sequences is well captured by the
Markov switching process. This is not surprising since the cross sectional leadership pattern of
each war type is captured by the stationary mixed strategy equilibrium. Because of Markov
switching, the chi-square distribution test is no longer applicable. However, the simulated p-values
(two-sided tests) suggest the null that the predicted frequency equals the observed one for 9 of the
10 leadership types cannot be rejected at normal significance levels. The null for one leadership
sequence is rejected at the 5% level, but not at the 1% level.
Table 7: Frequency Distribution of Leadership Types over 102 Wars
Leadership types
BP leads
Caltex leads
Shell leads
Observed
frequency
49
52
30
BP leads alone
27
Caltex leads alone
37
Shell leads alone
15
BP and Caltex lead
8
BP and Shell lead
8
Caltex and Shell lead
1
BP, Caltex and Shell
6
Total number of wars
102
Note: p-value is for two-sided test.
28
Predicted
frequency
47.8
54.8
29.8
27.3
33.4
13.7
11.6
6.3
7.2
2.6
102
p-value
0.83
0.43
0.96
0.95
0.31
0.77
0.23
0.58
0.013
0.11
Table 8: Predicted and Observed Number of Runs in the Leadership Sequences
Observed Runs
Predicted Runs
p-value
BP leads
Caltex leads
Shell leads
70
69
49
67.3
64.2
49.2
0.50
0.30
0.90
BP leads alone
Caltex leads alone
Shell leads alone
BP and Caltex lead
BP and Shell lead
Caltex and Shell lead
BP, Caltex, and Shell
51
71
30
17
16
3
9
48.5
62.5
27.3
19.5
13.1
13.5
5.9
0.59
0.09
0.59
0.78
0.49
0.015
0.46
Table 9: Predicted and Observed Cross Correlation Coefficients
Confidence interval
Correlation between
Observed Predicted
95%
99%
BP leading and Caltex leading
-0.43
-0.45
(-0.61, -0.28) (-0.66, -0.22)
BP leading and Shell leading
-0.02
-0.22
(-0.40, -0.04) (-0.42, 0.02)
Caltex leading and Shell leading
-0.36
-0.27
(-0.50, -0.09) (-0.44, -0.04)
Consider next the predicted and observed number of runs in each of the 10 leadership
sequences reported in table 8. The reported p-values are for the two-sided tests of the null that the
observed number of runs is the same as that predicted by the stochastic Markov switching process.
The null for 9 of the 10 sequences cannot be rejected at the 5% levels (the p-values are bigger than
0.30 in 8 of the 9 cases), and the null for none of the sequences is rejected at the 1% level.
Third, consider the predicted and observed cross correlation coefficients among the three
sequences of lBP , lCaltex , and lShell reported in table 9. The intuition behind the predicted negative
correlation coefficients among the three leadership sequences is simple. The estimated mixed
strategy equilibrium implies that when BP turns out to be a leader of a war, the probability that
Caltex or Shell also turns out to be a leader is small. On the other hand, when BP is not a leader,
the probability that Caltex or Shell is a leader is high. The observed correlation coefficients
between the BP and Caltex sequences and between the Caltex and Shell sequences are quite close
29
to the predicted values. The predicted correlation coefficient between the BP and Shell sequences
is within the 99% confidence interval of the predicted value as well.
Table 10: Predicted Transition Probabilities between the Four Types of Wars
BP leads alone previously
Caltex leads alone previously
Shell leads alone previously
Multiple firms lead previously
Probability that in the current war
BP leads Caltex leads Shell leads Multiple firms
alone
alone
alone
lead
0.12
0.38
0.28
0.21
0.58
0.07
0.10
0.25
0.11
0.68
0.02
0.19
0.12
0.41
0.07
0.40
Sum
1.00
1.00
1.00
1.00
Lastly, consider the predicted transition probabilities among the four types of wars reported
in table 10. The observed transition probabilities, which are very close to the predicted values, are
the estimated coefficients for regressions (4) through (7) reported in table 3. If a single firm led
alone in the previous war, the least likely transition possibility is that the next war is of the same
type. This is a direct result of the previous finding that a firm leads less often or plays relent with a
smaller probability if it led alone in the previous war. If multiple firms led together in the previous
war, there is a high probability (40%) that multiple firms will lead together again in this war. Even
if a single firm led alone in the previous war, there is at least a 19% chance that multiple firms will
lead simultaneously in this war.
6.4
Discussion and Additional Evidence
The results from the distribution test and the Markov switching process suggest that the
stochastic regularities in the observed leadership patterns are well captured by the presumed
equilibrium mixed strategies. The empirical results thus support the mixed strategy hypothesis.
If the seven leadership types all result from pure strategy equilibria, what explains both
the randomness in the outcome of the individual wars and the stochastic regularities of the
leadership outcome over the wars? In addition, pure strategy equilibria in wars of attrition
30
require at least one firm to immediately relent. This implies that the duration of each of the 102
wars should be exactly one day, which appears implausible. It is rather inconsistent with the
observation that a small number of wars appears to last longer than two or three days (see the
last war of attrition in figure 3 and Appendix B).
In fact, the presumed mixed strategy equilibrium has plausible predictions about the
duration of the wars of attrition. The probability that a war of attrition lasts exactly T days has a
geometric distribution, [ (1 − p A )(1 − pB )(1 − pC ) ]
T −1
[1 − (1 − pA )(1 − pB )(1 − pC )] .
Therefore, the
expected mean duration is
1/ [1 − (1 − p A )(1 − pB )(1 − pC ) ] ,
(15)
and the probability the war lasts T days or shorter is
1 − [ (1 − p A )(1 − pB )(1 − pC ) ] .
T
(16)
A war of attrition starts when the retail price is near or at the competitive level, but the start
date of the wars of attrition is not directly observable. For this reason, we cannot formally test
these war duration predictions. However, we do know a small number of wars appears to last
longer than three days and the duration of a war should be smaller than the duration of a price cycle
(because a war of attrition is only part of a price cycle). These observations allow for checking
whether or not the presumed equilibrium mixed strategies are reasonable from a perspective other
than the leadership outcome.
Table 11: Expected Duration of Wars of Attrition by Type
Mean duration (days)
Probability that duration is 1 day
Probability that duration is 2 days or shorter
Probability that duration is 3 days or shorter
BP alone
leads
1.99
0.50
0.75
0.88
31
In the previous war,
Caltex alone Shell alone
leads
leads
1.46
1.48
0.68
0.90
0.97
0.68
0.89
0.97
Multiple firms
lead
1.25
0.80
0.96
0.99
Table 11 reports the predicted mean duration of each type of war and the probabilities that
each war type lasts 1, 2, or 3 days or shorter. The predicted mean duration ranges from 1.25 days
and 1.99 days, indeed much smaller than the average length of the price cycles. The probabilities
in table 11 also suggest relenting should occur immediately in about 50% to 80% of the wars of
attrition and wars of attrition that last longer than three days should occur, but only infrequently.
7.
Conclusion
Mixed strategy is a fundamental concept in game theory, but is seldom tested in empirical
studies. A recent literature shows professional athletes play equilibrium mixed strategies in zerosum sports games. This paper tests equilibrium mixed strategies in a dynamic oligopoly market
setting before and under a unique simultaneous-move timing regulation. The strategic setting is the
war of attrition game at the bottom of retail gasoline price cycles that are well explained by the
Edgeworth price cycle equilibrium. The main finding is that the price leadership pattern observed
in the Perth market under the timing regulation, but not before, is characterized by the stationary
mixed strategies presumed by Maskin and Tirole. This finding supports the Bayesian view that a
player’s mixed strategy represents other players’ uncertainty of that player’s pure actions. This
finding also suggests agents are unlikely to play equilibrium mixed strategies if they can observe
and react to rivals’ action. However, in such situations, mixed strategy may still serve as a useful
technical device.
References
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Princeton University Press, Princeton.
33
BP
Caltex
Shell
34
Wholesale Price
03
/1
3/
01
05
/2
1/
01
02
/2
8/
01
02
/2
1/
01
02
/1
4/
01
02
/0
7/
01
01
/3
1/
01
01
/2
4/
01
01
/1
7/
01
01
/1
0/
01
01
/0
3/
01
03
/0
7/
01
95
Wholesale Price
05
/1
5/
01
05
/0
8/
01
90
Shell
05
/0
1/
01
85
Caltex
04
/2
4/
01
04
/1
7/
01
80
BP
04
/1
0/
01
04
/0
3/
01
03
/2
7/
01
03
/2
0/
01
03
/1
3/
01
75
80
85
90
Appendix A: Daily Brand Average Retail Price of Three Firms and the Wholesale Price
01/03/01—05/21/01
11
/0
1/
02
11
/1
5/
02
11
/2
9/
02
12
/1
3/
02
12
/2
7/
02
01
/1
0/
03
01
/2
4/
03
02
/0
7/
03
02
/2
1/
03
03
/0
7/
03
03
/2
1/
03
04
/0
4/
03
04
/1
8/
03
05
/0
2/
03
05
/1
6/
03
05
/3
0/
03
06
/1
3/
03
06
/2
7/
03
07
/1
1/
03
07
/2
5/
03
08
/0
8/
03
08
/2
2/
03
09
/0
5/
03
09
/1
9/
03
10
/0
3/
03
10
/1
7/
03
10
/3
1/
03
75
80
Prices: Australian cents per liter
85
90
95
100
105
12
/0
1/
01
12
/1
5/
01
12
/2
9/
01
01
/1
2/
02
01
/2
6/
02
02
/0
9/
02
02
/2
3/
02
03
/0
9/
02
03
/2
3/
02
04
/0
6/
02
04
/2
0/
02
05
/0
4/
02
05
/1
8/
02
06
/0
1/
02
06
/1
5/
02
06
/2
9/
02
07
/1
3/
02
07
/2
7/
02
08
/1
0/
02
08
/2
4/
02
09
/0
7/
02
09
/2
1/
02
10
/0
5/
02
10
/1
9/
02
10
/3
1/
02
75
80
Prices: Australian cents per liter
85
90
95
100
01
/0
3/
01
01
/1
7/
01
01
/3
1/
01
02
/1
4/
01
02
/2
8/
01
03
/1
4/
01
03
/2
8/
01
04
/1
1/
01
04
/2
5/
01
05
/0
9/
01
05
/2
3/
01
06
/0
6/
01
06
/2
0/
01
07
/0
4/
01
07
/1
8/
01
08
/0
1/
01
08
/1
5/
01
08
/2
9/
01
09
/1
2/
01
09
/2
6/
01
10
/1
0/
01
10
/2
4/
01
11
/0
7/
01
11
/2
1
11 /01
/3
0/
01
75
80
Prices: Australian cents per liter
85
90
95
100
105
Appendix B: Daily Market Average Retail Gasoline Price in Perth, 01/03/00—03/31/03
35