The Journal of Economic Dynamics and Control
Evolved Perception and Behaviour
in Oligopolies
Robert Marks,
Australian Graduate School of Management,
University of New South Wales,
Sydney NSW2052,
Australia
To appear in
The Journal of Economic
Dynamics and Control
Presented at
Computing in Economics and Finance,
Geneva,26–28June,1996.
Revised:May23,1997.
Please do not quote without the author’s agreement.Correspondence to Associate Professor Robert Marks,Australian Graduate School of Management,University of New South Wales,Sydney,NSW2052,Australia; E-mail:bobm@https://www.wendangku.net/doc/2f16743307.html,.au
ABSTRACT:
This paper builds on earlier studies which examined oligopolists in a repeated interaction as responding simply to past prices of their strategic rivals,and which used data from a mature market,with stable rules of thumb(mappings from past actions,or states of the market,to present prices)for the oligopolists’behaviour,whether purposefully learnt or emerging from the natural selection of the rivalry.The earlier studies imposed exogenous partitions on the action space,as perceived by the players.This study explores how such perceptions might be endogenised.A?rm answer to the question of how oligopolists partition their perceptions of others’actions,both through time and across the price space,will also provide information on how much or how little information they choose to use:in short,how boundedly rational the oligopolists have chosen to be.We use data from a retail coffee market to examine the evolved optimal partitioning and mapping of price space,manifest as the oligopolists’rules of thumb.The data suggest that brand managers are using very little information:whether prices changed or not.
ACKNOWLEDGEMENTS:
The author acknowledges many stimulating discussions with David Midgley, and his suggestion that Theil might have something to say about information and partitions;he also acknowledges Lee Cooper’s provision of the data used. An anonymous referee made helpful suggestions.Research support was provided by the Australian Graduate School of Management and the Australian Research Council.An earlier version was presented at the “Commerce,Complexity,and Evolution”conference,University of New South Wales,February,1996.
KEYWORDS:
repeated games,oligopoly,perceived partition,entropy,information.
1. Partitioning of Prices and History
This study follows from earlier work of the U.S.retail ground coffee market, in which we modelled players as responding simply to the past prices and other marketing actions of their strategic rivals(Marks Midgley&Cooper 1995;Midgley Marks&Cooper1997).That study used a genetic algorithm to derive“good”mappings from past actions to the marketing actions(such as price)to be pursued in the present period on the part of three strategic rivals. In the course of the earlier study we became aware of the importance of modelling not just the patterns of response of the strategic players,but also their perceptions,both in looking back and in discerning whether small price movements of their rivals’are strategically signi?cant.This study explores how such perceptions might be endogenised.A?rm answer to the question of how players partition1their perceptions of others’actions,both through time and across the price space,will also provide information on how much or how little information they choose to use:in short,how boundedly rational players are.
In the U.S.retail ground coffee market the price has been seen to vary from about$1.50per pound to about$3per pound.Cluster analysis shows that some prices and price regions are used more frequently than others for each brand,and so the earlier study used four of these for each brand as the four actions in its simulation.Even though our earlier studies constrained the arti?cially intelligent adaptive economic agents to these four actions(of the many they had been seen to use,and of the150or more that are feasible), we found that the historical pro?t performances could be improved using simple four-action,one-round-memory machines.
But cluster analysis is a crude technique.We wish to use the data to examine the price partitions that the players actually used.Such partitions will generally be in terms of price(and marketing action)levels,but the boundaries introduced mean that(away from the boundary)a one-cent-a-pound change in price is not a signal responded to by the other players,while that(at the boundary)such a small price change will be seen as strategically signi?cant by the rival players.It may be that we should partition the?rst differences of prices,so that a small change in price will not be perceived as a strategically signi?cant shift(no matter where the price was before the shift); only a price change(positive or negative,symmetrically?)will be seen as such.2
We?rst formalise the process that each player uses in deciding his action in the market from one week to the next,using a framework outlined _______________
1.The concept of partitioning in order to use the coarsest(or minimal)partition which is as
informative as the non-partitioned space was introduced by Blackwell&Girshick(1954), or earlier(Bohnenblust Shapley&Sherman1949).Radner(1972)describes a model in which there is no strategic uncertainty,and in which our partitions are his signals.
2.There is a literature on partitions as a means of imparting information to Bayesian-
rational agents(Geanakoplos1989,Samet1990).A recent paper(Dimitri1993)considers sequential experiments,in order to model certain information-processing skills on the part of agents.
by Lipman(1995).Each week,faced with the actual external state(or E-state),the players perceive an internal state(or P-state),which may update their beliefs,on which are conditioned their actions for the week,which together with the actions of their strategic rivals determine their pro?ts.
2. Formalities
There is a?nite set of external states(or E-states),?.The E-states are de?ned by the prices(and marketing actions)that each of the players charged for its brand for a large number of weeks into the past.?=A1×A2×A3, where A i is the vector of brand i’s prices(or actions)for all weeks into the past.
But it is unlikely that players perceive the information partition as ?nely as it is de?ned in the E-state.Nor is it likely that players remember more than a few weeks past in determining the internal stateΘ.There is a functionζ: ?→Θ,that tells which perceived P-state the player observes as a function of the E-state,whereζis the perception function:in E-stateω,the player observes P-stateζ(ω).
As a consequence,the true information content of the P-stateΘis thatωis one of the E-states generating this P-state:the true E-stateωis some element ofζ?1(Θ).If the P-state is optimally determined,then the lost information is valueless to the player—he or she is no worse off with the coarser partition of the P-state than with the?ner partition of the E-state. But if the P-state is sub-optimal,then the lost information is valuable,in that its use would result in a perception of the rivals’behaviour that would on average result in a higher pro?t for the player.
There will be a set of actions the player can choose from,denoted by A, with at least two elements.How or whether these actions are related to the perceived P-states is an empirical issue,although for the moment we shall use a distinct set.Note that since such perceptions are subjective,there is no guarantee that different players will perceive the same sets of P-states.
There will also be a pro?t function(usually in the form of a payoff matrix):u: A×?→R,which describes how the state affects the value of the different actions available to any player.Note that here?represents the true E-state of the market during the present week,and will not be known to the players until after they have each chosen their actions.Note,too,that players will only know their perceived states,not the true external states, even later.In general,one can assume a prior probability distribution q on ?,although in our case,discussed below,the probability distribution over external states is determined endogenously by the choices of the players in the market.
How does the P-stateΘdetermine beliefs about the external state?Let ?denote the set of probability distributions on?.Thenβ: Θ→?is the belief function.Beliefs matter because actions are contingent on them.The mapping from belief to actionα: ?→A is the action function.
Figure1illustrates the model from external E-states to?nal payoff, with an example of an E-state,showing how it is transformed into a P-state and an action.As Lipman points out,it is possible to combine perception,
E-states?
ω={1.59,2.40,2.27;2.00,1.75,2.30;...}
partition functionζ
P-statesΘ
θ={low,High,high}
belief functionβ
Beliefs?
δ
action functionα
Actions A
a={Low}
payoff function u
Payoffs
{857.3}
Figure1:From External State to Payoff:The Player Modelled
belief,and action functions into a behaviour rule:f: ?→A,where f(ω) =α(β(ζ(ω))).This corresponds in the?gure to an arc from the E-states node to the Actions node.In our earlier studies(Marks et al.1995;Midgley et al.1997),the partitioning mapping from the E-states node to the P-states node was exogenously determined,and we used a Genetic Algorithm(Mitchell 1996)to search for arcs(mappings)from the P-states node to the Actions node,using the payoffs as?tnesses.
We can check the internal consistency of the belief functionβ.No processing:δ=β(θ) ?θ,which suggestsδ=q,the prior distribution.Full processing:θ≠θ′?β(θ) ≠β(θ′).We expectβ(θ)to put probability1on the setζ?1(θ).As Lipman puts it,the player should be able to say to himself:“My beliefs areδ.But I know I’d have these beliefs if and only ifω∈W.So I shouldn’t be putting any probability on states outside W.”
Lipman distinguishes between interim optimality and ex-ante optimality.For the former,an action functionα: α(δ)which maximises the following function for allδmust be derived:
Σu(a,ω)δ(1)
ω∈?
Then a behaviour rule must be constructed by letting f(ω)equal the action α(δ)where P-stateζ(ω)results in actionδ.That is,for eachω:β(ζ(ω)) =δ, let f(ω) =α(δ).This describes how the player will behave in searching any given solution.
Ifβ(ζ(ω))=β(ζ(ω′)),then f(ω) =f(ω′).If the player has the same beliefs in two E-states,then his behaviour is the same in those E-states;that is,f is measurable with respect toβ(ζ)).
In our earlier studies,we derived equation(1),and we used evolutionary ?tness to proxy for u.Our action function wasα(δ),whereδis the player’s belief of the E-state:we explicitly separated the determination ofδand the determination ofα.
The belief functionβ: Θ→?can also be endogenous:if f is an interim-optimal behaviour rule f: ?→A,then let V be the ex-ante expected pro?t associated with processing information according to the functionβ:
Σu(f(ω),ω)q(ω).
V(β)=
ω∈?
Recall thatβaffects f by imposing a constraint on the set of available f s via the measurability requirement.Then we can model the determination ofβby supposing that the player chooses it(endogenously)so as to maximise V(β) ?c(β),where c gives the expected information processing costs. Typically,the player is constrained to chooseβfrom a set B of belief functions.
Lipman raises the question:Is it odd to model bounded rationality by assuming optimal information processing?Why not just choose optimally a givenω?Well,we assume general knowledge,that is,how to solve,not the speci?c solution.The model shows how to chooseβand f contingent onω. Moreover,if players do not achieve optimalβand f,then the model of the world as the player sees it is not completely speci?ed.
3. Partition Models
Information processing can be summarised by a partitionΠof the set of E-
states?.A partitionΠof a set?is a collection of subsets of?with the
property that everyω∈?is in exactly one of these subsets.The elements of
the partitionΠare often referred to as events.Intuitively,a partitionΠis
said to be?ner than a partitionΠ′when learning which event ofΠcontains a
givenωconveys more information than learning only which event of?′
containsω;the converse is a coarser partition.
Partitions are closely related to equivalence relations:binary
relationships that are re?exive,symmetric,and transitive.Given any
equivalence relation R(such as two E-statesωandω′are equivalent if they
lead to the same beliefs),we can partition?into equivalence classes:for any
givenω∈?,let
R(ω)=(ω′∈?|ωRω′).
Thus R(ω)is the set of points equivalent under R toω—the equivalence
classes of R—which it is readily shown form a partition,?,called the
partition induced by R.Similarly,given any partitionΠ,we can de?ne an
equivalence relation induced by the partition by saying thatωandω′are
equivalent if they are contained in the same event inΠ.
Consider the equivalence relation over?de?ned by saying that two E-
statesωandω′are equivalent if they lead to the same beliefs(or if β(ζ(ω)) =β(ζ(ω′))).LetΠdenote the partition induced by this equivalence relation,and letπ(ω)denote the event ofΠcontainingω.
The key to the partitional models is that beliefs are assumed to be
internally consistent,so that if the player’s beliefs areδonly in certain
external E-states,thenδmust rule out any other https://www.wendangku.net/doc/2f16743307.html,ually,the
stronger assumption is made:thatδis assumed to be calculated from the
prior q via Bayes’Rule.The partition can be used to summarise the player’s
information processing without explicit reference to the underlying belief
function.
The partitionΠis easily interpreted in terms of information processing:
ifΠhas only one event(the entire set?),then the player is not processing his
input P-states at all,which corresponds to the case whereβ(θ) =q for every
P-stateθ.By contrast,a partition that has a different event for each different
E-stateωinvolves complete processing:the player processes the information
so thoroughly that he recognises every possible distinction between inputs.
His partition could not be?ner.
BecauseΠsummarises information processing,write V(Π)instead of
V(β),for the expected pro?t associated with information processing according
to the belief functionβ,which is identical to the expected pro?t associated
with the information partitionΠ.If we further assume that the cost of a
given information processing functionβdepends only on the partitionβ
generates,then we can work with c(Π)instead of c(β)for the expected
information processing costs.
4. Finite automata
Aumann(1981),Neyman(1985),and Rubinstein(1986)were the?rst to study repeated games in which players were restricted to using?nite automata to implement their strategies.One reason for studying game-playing machines is that they can be used to give a formal description of the concept of“bounded rationality”(Simon1972).Finite machines must by de?nition be bounded,and can be used to model the concept.Neyman and Rubinstein modelled bounded rationality as limitations on the number of states of the machine:Neyman imposed an exogenous limit on the number of states;Rubinstein assumed a cost trade-off.
An automaton consists of a number of internal states,one of which is designated the initial state;a transition function,which speci?es how the automaton changes states in response to the other players’actions;and an output function,which maps state to action.See Marks(1992)for a fuller treatment.In our earlier studies,(Marks et al.1995;Midgley et al.1997)we used the genetic algorithm to determine the initial state and the mapping from state to action.
Let I denote the set of possible histories of play(of actions).Then with three players I=A1×A2×A3,where A i is the history of player i’s actions in the game.A strategy in the game is a functionσthat speci?es an action as a function of the state of the game,which in turn is a function of the history of the game.If the game has an unlimited number of rounds,then after any history h the remaining game is still in?nite.Hence a strategy for the overall game,σ,speci?es a continuation strategy following h for the game.Kalai& Stanford(1988)call this the induced strategy,σ|h.We can say that two histories,h and h′,are equivalent underσif they lead to the same induced strategy;σ|h=σ|h′.
Lipman argues that it is easy to show that this is an equivalence relation,so that it generates a partition of the history set I,which can be denoted by I(σ).If the player knows which event of this partition a history lies in,then he knows enough about the history to determine the strategy it induces.Kalai&Stanford show that the number of internal states of the smallest automaton which plays a given strategy is equal to the number of sets in this partition,when the“Moore machine”representation is used (Moore1956).Banks&Sundaram(1990)considered the number of states and the number of transitions,in a two-dimensional measure.
In our earlier studies,the set of external states?is the set of histories I =A1×A2×A3,where we model the strategic interaction of three brand managers as players,following Fader&Hauser’s1988study.We arbitrarily chose a time partition of one-round memory,so that no actions of more than a week ago were directly perceived by the players(although indirect in?uences through others’actions last week were not,of course,excluded).To partition the large number of possible prices,we used a statistical technique on historical data of the oligopoly,namely cluster analysis,in order to partition the price space into four bands,again an arbitrarily chosen number.The boundary prices varied with brand.
These techniques allowed us to map the E-state of brands’prices(and other marketing actions)for many weeks into a much coarser P-state of one week’s data,suitably partitioned;an exogenous perception functionζ: ?→Θ.As described,we then used the machine-learning genetic algorithm to search for better mappings from P-state to action,orα(β(θ)).Note that,using machine representations,we did not explicitly model beliefsδ,or a belief function(β: Θ→?),or how actions are mapped from beliefs(α: ?→A). Instead,we can de?ne our response function as a mapping from P-state to action:γ: Θ→A.See our earlier studies(Marks et al.1995;Midgley et al. 1997)for discussion of this search.
The set of actions,A,is the set of strategies for the repeated game. Hence,following Lipman,any strategyσcan be described as a behaviour rule f from I(σ)into A,where f(h) =σ|h.Thus we can separate the choice of a strategyσinto the choice,?rst,of a partition on the set of historiesΠ,and, second,of a function fromΠto the set of strategies or actions.3The cost function c is usually taken as an increasing function of the number of events of the partition only,c(Π),although other functions are possible.
A related model is Dow’s1991model of search with limited memory. The E-state is a pair of prices,one observed in period1and one in period2. The action is which price at which to buy.Dow’s agent knows(or believes)he will not be able to remember the?rst price exactly(an exogenous constraint) and so is modelled as partitioning the set of possible prices;his memory is only into which event of the partition the period1price fell.The partition is explicit,and Dow also assumes that costs are proportional to the number of events in the partition.
The motivation for this study is the desire to endogenise the partitioning which is necessary for simulations of interactions as mappings from state of the market,suitably de?ned,and actions by the players.Although this study does not go beyond the discussion below of how to partition to maintain maximal information,future work will harness these results in further simulations,as discussed above.
5. Optimal Partitioning
Lipman discusses a class of models in which although the E-state is observed directly,it is classi?ed according to which of two sets it falls:whether or not it is above a certain real-valued threshold.4There seems no reason why the concept should not be generalised to multiple thresholds.The exogenous partitioning of our earlier studies was into four regions,requiring three thresholds,but we have considered a?ner partition.Although the _______________
3.Lipman(1995,fn.5)points out that not every partition of I can be generated by some
strategy,and that not every function from such a partition to A will constitute a legitimate strategy.
4.If the E-state is not already expressed as a real number,it must?rst be translated into a
real number.In our case,however,prices are real numbers,up to the integers.
programming effort increases in the number of thresholds,Lipman reports that this class of models assumes zero cost for information processing.
5.1 A First Cut
We start by considering the simplest partition of the price space,into two regions,a dichotomous partition between“low”and“high”prices.The question is where best to draw the boundary between the two regions.To explore this issue,we set up a model in which the choice of where to divide the region between the lowest price and the highest price is one of eight points,dividing the price space into nine equal regions.The data are78 weeks of weekly observations for three competing brands in a mature market (that for canned,ground coffee in a U.S.city).5
From above,the set of external states?of the market with three strategic players is the set of histories I=A1×A2×A3,but we wish to de?ne a new set of market states based on the perceived statesΘ.Instead of the set of E-state histories I,de?ne a set of histories I?i=A?1i×A?2i×A?3i,where A?ji is the history of actions of player j as perceived by player i.As soon as we introduce subjective perceptions into the game,we introduce the possibility of subjective histories,too,but,so long as the partitioning which gives rise to the perceived actions of self and others is endogenous,no player could improve his or her payoffs by changing his or her partitioning of the price space,at least in equilibrium.From a learning or evolutionary viewpoint, players will adjust their perceptions(their partitioning)so as to end up close to their notional equilibrium partitioning.
We also consider dichotomous partitioning of the?rst differences of price (both absolute and algebraic)in order to model players’responses to price jumps,as well as considering a symmetric terchotomous partition of price levels.
5.2 Measures of Optimality
Which partitioning is best?
We argue that the best partition is the one that loses the least amount of information.One candidate is the partition(or partitions)which result in the highest number of perceived states,but there is a more informative measure:Theil(1981),in discussing the general issue of information measures associated with events,suggests entropy.6Entropy H is given by
_______________
5.For further details,see Midgley Marks&Cooper(1997).The three rivalrous brands are
Folgers,Maxwell House,and Chock Full O Nuts.
6.As discussed in Section7below,information is merely the means to an end:the player’s
pro?ts,or expected pro?ts in a stochastic game.But,as McGuire(1972)argues,the search for a one-dimensional measure of“informativeness”—the value of a“information structure”or partition—is in vain;entropy included.See also Radner(1987,p.300):“...there is no numerical measure of quantity of information that can rank all information structures[partitions]in order of value,independent of the decision problem in which the information is used”.Note that Kolmogorov introduced the concept of the entropy of a countable partition in1958(Iyanaga&Kawada1977).
H ≡?i =0ΣN ?1
p i log b p i ,(2)
where there are N perceived states,and the probability (or observed frequency)of state i is p i .If the logarithm base b is 2,then the units of entropy are “bits”;if b is the exponential constant e ,then “nits”.
Theil argues on axiomatic grounds that entropy is justi?ed,by showing that the entropy information measure of an event (such as the observation of a speci?c market state)satis?es four axioms:
Axiom 1.The information content of observing that a state i occurred depends
only on the probability p i of its occurrence prior to the observation.
Axiom 2.The information is a continuous function of p i in 0
monotonically decreasing.
Axiom 3.When state i is certain,its observation carries zero information.
Axiom 4.The information content of a state which is the union of two
mutually exclusive states (zero intersection)equals the sum of the information of observing one state and that of observing the other.(Additivity.)
The maximum number of states is equivalent to entropy as an information measure only when each state is equally likely or frequent as is readily seen in equation (2)with p i = 1/N , ? i .With non-uniform distribution of states,the measure of the maximum number of states N throws away information about each state’s frequency.None the less the two measures are empirically close at determining the optimal partition point with dichotomous partitioning.In order to better compare the two measures,we use the antilogarithm of entropy,or alog entropy (AE ),which is given by the expression:
AE ≡antilog b H ≡b H =i =0ΠN ?1p i
p i 1_______,(3)
where b is the base of the logarithm used in equation (2).This measure,unlike entropy,has the additional bene?t of being independent of the base b .The units of the measure of alog entropy are “equivalent states”.
A dichotomous partition divides the price line into two regions only:“low”(below some partition point λ)and high”(above it);there remains the empirical issue of the optimal location of the dichotomous partition point.Since there is only one degree of freedom in its choice,we can plot any measure against its location.
Figure 2uses the market data from Chain One 7to plot two measures of information losses against the position of a dichotomous partition,as it moves in steps of a thousandth of the range between lowest price and highest price charged by each of the three brands over the 78-week period of the data.The
measures are:
1.The number of perceived states;and
2.The closely related measure of sample alog entropy across all perceived
states,from equation(3).
The two measures are brand-or player-independent,since they don’t require consideration of the actions that result from the perceived states,by player.
The steps we follow are:
1.For each of the three brands or players,determine the minimum and
maximum prices charged over the period.
2.For the following?ve steps,choose a partition pointλ,(0 ≤λ≤1).With
iteration of these steps,the partition point will increment from zero to one in steps of a thousandth of the range between each brand’s minimum and maximum prices.8
3.For each brand’s price,for each week,for a given partition pointλ,
determine whether the price is“low”or“high”:if
price≤minimum price+λ(maximum price–minimum price), then the price falls“low”,otherwise“high”.Whenλ=0,almost all prices are classi?ed as“high”,since the partition point is at the minimum price,and so the“low”set is almost empty;whenλ=1,all are classi?ed as“low”,for the opposite reason.(Note that the minimum and maximum prices in the expression are brand-or player-speci?c,since there is no constraint on players to price in the same range as do their rivals.)
4.For the given partition point,determine the perceived state of the
market.With three players and one-week memory,there are23or8 possible states.Arbitrarily number them by calculating the state number:
state=4×F+ 2×MH+CFON,
where F is Folgers’action,MH Maxwell House’s,and CFON Chock Full O Nuts’;if a player priced“low”last week,then de?ne that player’s action to be0,otherwise1.(For Figure4,with two-week memory,there are26or64possible states,similarly arbitrarily numbered.)
5.For the given partition point,calculate the observed frequency p i of each
state i,as de?ned by the number of times each perceived state is observed as a proportion of the total number of times states are perceived.
_______________
7.We had scanner data from three supermarket chains,but here use data from Chain One
and(in Figure3)Chain Two only.
8.With the data we have,increments smaller than one thousandth do not reveal any?ner
structures in the entropy measure:one thousandth is a suf?ciently?ne increment.
6.For the given partition point,these frequencies can be thought of the
sample probabilities of the perceived states,which enables calculation of the alog entropy measure associated with a particular partition point, from equation(3).
7.As mentioned above,the the total number of perceived states would be
equal to the alog entropy if the frequencies of all perceived states were equal:using the total number of perceived states as a measure of the effectiveness of a partition point at retaining relevant information means throwing away the information of the frequencies of the perceived states,for the given partition point.Alog entropy is bounded above by the the total number of perceived states,as seen for instance in Figure2.
8.After incrementing the partition point by one thousandth,Steps2to7
are repeated,until the full interval between minimum and maximum prices has been searched,as plotted in the Figures below.
We examine the possibility of price changes,as well as price levels, below,calculating both measures.9
6. Results
Following the eight-step procedure listed above,we use78weeks of data from Chain One to plot the measures in Figure2.This is compared with Chain Two data in Figure3.For the Chain One data,we have also considered two-week memory(Figure4);?rst-differences in prices with one-week memory, both absolute,in which only the size of the price jump matters(Figure5),and algebraic,in which both size and direction of the price jump matter(Figure 6);and a symmetrical terchotomous partition,in which the price range of each brand is divided into three partitions,symmetrically about the mid-point of the range,which requires a single parameter only(Figure7).
The six cases are summarised in Table1and the results are summarised in Table2,below.
_______________
9.An earlier version of this paper explored the possibility of a third,brand-dependent
measure.This was based on the concept of the mappings from state to action of a speci?c brand manager.It is possible to derive a matrix,the rows of which correspond to perceived states,given the partition point,and the columns of which correspond to a?xed number of price ranges which span the actions of a speci?c player.Intuitively,when the partition is such as to maximise the number of perceived states,the best partition is that which minimises the mean mapping from perceived state to action.The rationale is that, under the ideal partition,all possible states are perceived,and that each state is found to map to only one action in the historical data.If,in the limit,it is found impossible to reduce the number of actions per perceived state to one,with all states perceived,then this may be due to one of several possibilities:a misspeci?cation of the model(it may be, for instance,that players respond to not price levels but to price changes),or that the assumption of a deterministic mapping from state to action is wrong,with some mixing of strategies.We do not explore this further here.
Figure Data Partitions Weeks of Price
(Chain)Memory Variable
_______________________________________________________
2One21level
3Two21level
4One22level
5One21|difference|
6One21difference
7One31level
_______________________________________________________
Table1:Summary of the Cases.
6.1 Dichotomous Partition of Price Levels
6.1.1 One-week memory.Examination of Figure2reveals that entropy is a much?ner measure than is the number of perceived states:over ranges of the partition pointλ,the latter measure is unchanging,while the entropy varies. Moreover,from Figure2the entropy measure suggests that the optimal dichotomous partition for Chain One data is a thresholdλat90.1%This means that for a dichotomous partition with the observed behaviour of the three strategic brands in Chain One,less information is lost with a partition point at90.1%of the distance between each brand’s minimum and maximum prices than with any other partition point.
For Chain Two data,the optimal partition point using the entropy measure is signi?cantly lower,at between71.4%and71.5%,as seen in Figure https://www.wendangku.net/doc/2f16743307.html,paring Figures2and3for the two sets of data,we can characterise the players in Chain Two as more responsive to prices in the mid-upper range than the players in Chain One appear to be.In Chain One,the threshold between“high”and“low”prices is around90%of the price range between highest and lowest price of each player;in Chain Two it is around71%.
6.1.2 Two-week memory.Players with two-week memories can respond to movements in other players’prices,unlike players with one-week memories only:a rise(from“low”to“high”),a fall,a steady“low”,or a steady“high”on the part of each of the other players(as well oneself)can be responded to. With two-week memory,three players,and dichotomous partitioning,there are26=64possible states of the https://www.wendangku.net/doc/2f16743307.html,parison of Figures2and4 reveals a richer information structure of the same data set(Chain One).But the entropy-maximising partition pointλis exactly the same:at90.1% precisely(using steps of0.001of the range).This is only an artefact of the data,as is seen when the Chain Two data are analysed with a two-week memory model:the maximum entropy partition is slightly higher than in Figure3,at72.4%.
6.2 Dichotomous Partition of First Differences of Price
Using two-week memory the model can track the directions of players’price movements,but not,with dichotomous partitioning,the magnitudes of price
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jumps.For this reason,and since it may better describe the way the historical brand managers behaved,we consider not price levels,but the?rst differences of prices:price at week t minus price a week earlier.
Figure5deals with absolute?rst differences,so that only the magnitude but not the direction of the price jump matters,and Figure6deals with the actual(or algebraic)?rst difference,which include both magnitude and sign of the change.For Figure5,the partition point is between0%and100%of the range between the smallest(greater than or equal to zero)and largest absolute?rst difference in price for each player;for Figure6,between0%and 100%of the range between the smallest(most negative)and largest(most positive)?rst difference for each player.
6.2.1 Absolute First Differences.From Figure5,we see that entropy is maximised atλ=1.4%and1.5%(with steps of0.001),which means that with a dichotomous partition the most signi?cant threshold is whether or not the absolute value of a price change is greater than or less than about1.4%of the range between smallest absolute price change and largest.Intuitively,the threshold is the boundary between no change(or negligible change)in price from one week to the next,and a signi?cant change(which is here any absolute change greater than1.6%,up or down).
6.2.2 Algebraic First Differences.Figure6is plotted for actual?rst differences in price.As we might expect,it is more closely symmetrical than is Figure5,since price rises and falls register distinctly here.Maximum entropy occurs at aλequal to the exact midpoint of the range of price differences.This seems consistent with the results of Figure5,but there is no reason,ex ante,to believe that price rises and falls should be even roughly symmetric:a pattern of small falls followed by large rises—such as is sometimes seen in petrol price wars(Slade1992)—will bias the data,and the midpoint will correspond to a positive price jump,but such asymmetries are not seen in these data.
6.3 Symmetrical Terchotomous Partition of Price Levels
A dichotomous partition is the simplest we can consider(and the easiest to calculate:above or below the partition point);with only one degree of freedom,it is also the easiest to plot,as in Figures2through6.But it may be that players use more sophisticated partitioning of the price space.For terchotomous(3)or higher-order partitions,there are more than one degrees of freedom,in general,which is more dif?cult to search for and not as easy to present graphically.There is,however,one way to model terchotomous partitioning using only one degree of freedom:λis the proportion of the range spanned by the central partition of the three,centred on50%,so thatλ=0 corresponds to two partition points together at the centre point of the price range;λ=1corresponds to one partition point at the bottom(left)of the price range and the other at the top(right);λ=50.0%corresponds to one partition point at the quarter point and the other at the three-quarters point of the price range—hence the description symmetrical terchotomy.
Figure7presents the results of using this partitioning with the Chain One data.The maximum entropy thresholdλoccurs at80.3%.This means
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