Volume 35, Issue 4
A Rationalization of Selecting Behavior to Release the Cheap Edition
Yuhki Hosoya
Department of Economics, Kanto-Gakuin University.

Abstract
This article examines a specific sales technique employed by certain product suppliers (such as book or gaming
software suppliers). Such suppliers release cheap editions (e.g., paperback versions following the initial hardcoveredition) of only those products that had high sales volumes and not those that had low sales volumes. This behavior
seems counterintuitive because the additional cost of producing such cheap editions is approximately zero (it is asignificantly lower cost than releasing a new product) and the cheap editions will generate a profit even if they do notsell well. Hence, it would seem rational for a supplier to release a cheap edition of each of their products following theinitial release. This article explains this behavior, showing that it is not necessarily irrational. We construct a differential
equation comparable to the replicator dynamics of a strategic form game, and show that if a supplier's decision isbiased then their payoff may be better than if it is not biased.

We are grateful for worthful comments and suggestions of anonymous referees.
Citation: Yuhki Hosoya, (2015) ''A Rationalization of Selecting Behavior to Release the Cheap Edition'', Economics Bulletin, Volume 35,
Issue 4, pages 2617-2624
Contact: Yuhki Hosoya - hosoya@kanto-gakuin.ac.jp.
Submitted: July 14, 2015. Published: December 13, 2015.

1

Introduction

This article rationalizes a behavior specific to certain book and software
suppliers: releasing cheap editions of only such products (e.g., paperbackbooks and cheap gaming software versions) that sold well on their initial
release. The cost of releasing the cheap edition tends to be much lower thanof creating a new product and we can consider its cost to be approximately
zero: the suppliers thereby achieve a profit even if this cheap edition does
not sell well. Releasing cheap versions of only those products that achieved
high sales volumes seems to contradict the profit-maximization objective ofsuppliers; rationally, they should release a cheap edition of each product. It
can be argued that there is a lower motivation to release a cheap editionof a product with high sales than of a product with low sales because thelatter product has not reached a relatively large potential market. Althoughsomewhat persuasive, this argument contradicts the actual supplier behavior.
This article explains and clarifies such supplier behavior, showing that
it is not necessarily irrational. Specifically, if the suppliers release a cheap
edition of every product, then most consumers will wait for this version andwill tend not to buy the initial release. To avoid such waiting behavior, thesupplier does not release a cheap edition when their product has poor sales:
the suppliers aim for long run, rather than short run, profits.
We use a dynamic system that is similar to the replicator dynamics ofa strategic form game1 to represent this situation. In this system, the supplier’s behavior is biased and they tend not to release a cheap edition of theirproduct when many consumers are waiting for it. We show that if this bias
is sufficiently high, then the supplier’s payoff may be better than if their bias
is zero.
This method is of value, particularly compared with alternative explanations (especially, those that use the Folk theorem). First, it does not require
any clever actions by the consumers. Second, the supplier’s decision to realize the long run equilibrium emerges specifically and it matches the actual
behavior.
In section 2, we present our model and derive its results. The concluding
remarks are given in section 3. The proof is in the appendix.

1

The replicator dynamics was introduced by Taylor and Jonker (1978), and named
by Schuster and Sigmund (1983). They consider a symmetric situation. However, thereplicator dynamics used in this research is an asymmetric case, which was treated by
Selten (1980). See Hofbauer and Sigmund (1998) or Weibull (1997).

2

Model and Result

2.1

Replicator Dynamics

Consider the following strategic form game.

ABC(a1 , b1 )(a2 , b2 )D
(a3 , b3 )(a4 , b4 )Let (x, 1 − x) (resp. (y, 1 − y)) be the mixed strategy of the above gamein player 1 (resp. player 2), and consider the following dynamic system.
ẋ/x = (1 − x)[y(a1 − a2 ) + (1 − y)(a3 − a4 )]ẏ/y = (1 − y)[x(b1 − b3 ) + (1 − x)(b2 − b4 )].
This equation is called the replicator dynamics, which can be considered
as a process representing players’ adjustive behavior. Let us explain thisdynamics. Consider that x is the proportion of type 1 agents such that theychoose A, and that y is the proportion of type 2 agents such that they choose
C. At first, all players blindly believe that their current action is the best.
As time goes by, however, they may observe that their action is not optimaland they may change it. The change occurs in proportion to the difference
between the payoff of the current action and of the social average. Therefore,
the growth rates of both proportions x and y are equal to this difference, andthe above equation appears.
Selten (1980) showed that, in general, only the pure Nash equilibrium(NE) may become the stable steady state in such an asymmetric replicator
dynamics.

2.2

The Model

Consider a market that consists of two players. Player 1 represents a consumer and player 2 represents a supplier. Player 1 considers whether they
will immediately purchase a product on its initial release. If they do, thenthey gain u > 0; otherwise they wait for the release of the cheap edition ofthis product. If player 2 releases a cheap edition, then player 1 can buy it andgain d > 0. If player 2 does not release a cheap edition, then player 1 waits
until this product becomes dated and its worth tends to zero. The product’s
original retail price is denoted by pH and pL denotes the cheap edition price.
We assume pH > pL > 0.

If u − pH ≥ d − pL , then the consumer should immediately purchasethe product. This consumer must not consider a cheap edition release in
their purchase decision, and thus we restrict our consideration to the casein which d − pL > u − pH . Moreover, if u − pH ≤ 0, then the consumer
should wait for the cheap edition. This consumer behavior is simple and wetreat it implicitly. We assume that if player 2 does not release the cheap
edition, then they automatically lose D > 0, (the sales for such consumers).
Therefore, we consider only the case where d − pL > u − pH > 0.
The following strategic form game represents the above situation. Strategy B represents “Buy this product immediately” and strategy NB represents“Not buy until the supplier releases the cheap edition”. Similarly, strategy
R represents “Release the cheap edition” and strategy NR represents “Notrelease the cheap edition”.

B
NB

R(u − pH , pH )(d − pL , pL )NR(u − pH , pH − D)
(0, −D)

Clearly, the unique NE of this game is (N B, R). Actually, we see that
R is the dominant strategy for player 2. We call the payoff pL of player 2
in this NE a short run profit for the supplier. In NE, the supplier must
release the cheap edition and the consumer must wait for this cheap edition:
as in the introduction, there is a mismatch between this NE and the actual
behavior of suppliers.
Next, let (x, 1 − x) (resp. (y, 1 − y)) be the mixed strategy for the above
game of player 1 (resp. player 2). Consider the following dynamic system:ẋ/x = (1 − x)[(u − pH ) − y(d − pL )],ẏ/y = (1 − y)[(1 − x)pL + D] − b(1 − x).

(1)

If a parameter b (indicating supplier’s bias) is zero, then this dynamic systemis the same as the replicator dynamics of this game. However, in such a case,the unique NE of the game is globally stable. Therefore, a mismatch between
theory and the actual behavior of suppliers still remains.
Therefore, we add the term −b(1 − x) to represent the supplier’s decisionbias. This term represents the supplier behavior of “not releasing the cheap
edition if the initial product sales x are not sufficiently high”. Note thatthe existence of a positive b is the observed fact: in the real world,many suppliers do not release a cheap edition unless the initial product sales
are sufficiently high. However, if b > 0, then the unique NE is not thesteady state of equation (1), and it may seem that the existence of positiveb indicates that the supplier is irrational.
We define the long run profit as the supplier’s payoff for the mixed
strategy profile ((x∗ , 1 − x∗ ), (y ∗ , 1 − y ∗ )), where (x∗ , y ∗ ) is the asymptoticallystable steady state of equation (1).2 Our purpose is to show that there existsb > 0 such that the long run profit is greater than the short run profit
pL , which indicates that the existence of a positive b does not mean theirrationality of the supplier.3
In fact, the following proposition shows the existence of such a b.
L )−(u−pH )]Proposition: If b > (pH +D)[(d−p, then this dynamic system has an
d−pL
asymptotically stable steady state such that,(x∗ , y ∗ ) = ((d − pL )(pL + D − b) − (u − pH )(pH + D) u − pH,).
(d − pL )(pL − b) − (u − pH )pHd − pL(pH −pL )(d−pL )Moreover, if D < (d−pand b > 0 is sufficiently high, then the longL )−(u−pH )run payoff exceeds pL .

Proof : See appendix.

3

Concluding Remarks

This article explained and clarified the specific supplier (of books or gaming
software) behavior of releasing a cheap edition of only such products thatsell well. We showed that if the supplier’s short run decision is biased, then
their long run payoff may improve. Therefore, despite its appearance suchsupplier behavior is not necessarily irrational and may actually improve their
payoff.
We used a dynamic system similar to the replicator dynamics of a strategic form game to represent this situation, including a biased supplier behav2

The terms long run and short run reflect our interpretation of the model. We thinkthat if suppliers only consider today’s profit, then they must choose the dominant strategy
R. Therefore, if their behavior against R can be justified, then they must consider the
profit in the future.
3
The left-hand side of (1) is ẏ/y for technical reasons only. If we rewrite (1) asẏ = y(1 − y)[(1 − x)pL + D] − b(1 − x),then y may become to be negative because the effect of the term −b(1 − x) is not restricted, which contradicts the interpretation of y. Later our lemma 1 will show that sucha problematic case does not occur for the solution of (1).

ior. This method is of value, particularly compared with alternative explanations, especially using the Folk theorem. First, it does not require any cleveractions by the consumers. Second, the supplier’s decision to realize the longrun equilibrium emerges specifically and matches the actual behavior.

A

Appendix: Proof of Proposition

First, we will prove the following lemma:
Lemma 1. Consider equation (1) with an initial value condition (x(t∗ ), y(t∗ )) =(c1 , c2 ) ∈]0, 1[2 , where b ≥ 0. Then, the solution (x(t), y(t)) can be extended
on [t∗ , +∞[ and (x(t), y(t)) ∈]0, 1[2 for every t ≥ t∗ .
This lemma is needed for assuring the following two properties. First, thestability of the equation has a meaning, because the solution (x(t), y(t)) canbe defined for an infinitely long period. Second, the value of (x(t), y(t)) is
consistent with the interpretation of these: if (x(t), y(t)) ∈/ [0, 1]2 , then thiscannot be treated as a mixed strategy.
Proof. Consider the following modified dynamic system of (1):ẋ = x(1 − x)[(u − pH ) − y(d − pL )],ẏ = y[(1 − y)[(1 − x)pL + D] − b(1 − x)],(2)where the right-hand side is defined on R2 . Clearly, any solution (x(t), y(t))
of (2) with x(t) > 0, y(t) > 0 for any t is also a solution of (1). Therefore,
it suffices to show that the solution (x(t), y(t)) of (2) can be extended on[t∗ , +∞[ and (x(t), y(t)) ∈]0, 1[2 for every t ≥ t∗ . If (x(t), y(t)) cannot be
extended on [t∗ , +∞[, then there exists t̂ such that (x(t), y(t)) ∈/ [0, 1]2 for4
2t > t̂. Therefore, it suffices to show that (x(t), y(t)) ∈]0, 1[ for all t in thedomain of (x(t), y(t)) with t ≥ t∗ .
Suppose that x(t) ≤ 0 for some t > t∗ . Let t+ = min{t′ ∈ [t∗ , t]|x(t′ ) = 0},
and consider the following equationż = z[(1 − z)[pL + D] − b], z(t+ ) = y(t+ ).
Choose any solution z(t′ ). Then, (0, z(t′ )) is a solution of (2) defined ona neighborhood of t+ . Because of the local uniqueness of the solution, we
4

It is well known that for any nonextendable solution z(t) of an autonomous ordinary
differential equation ż = f (z) defined on ]a, b[, either b = +∞ or for any compact set C
included in the domain of f , there exists t ∈]a, b[ such that z(t′ ) ∈/ C for any t′ ∈ [t, b[.
See Pontryagin (1962) for more detailed arguments.

have x ≡ 0 on a neighborhood of t+ , which contradicts the definition of t+ .
Therefore, we have x(t) > 0 for all t > t∗ . By the same arguments, we canverify that x(t) < 1 and y(t) > 0 for all t > t∗ , and if b = 0, then y(t) < 1
for all t > t∗ .
Next, suppose that b > 0 and y(t) ≥ 1 for some t > t∗ . Let t+ = min{t′ ∈[t∗ , t]|y(t) = 1}. Because y(t∗ ) < 1, we have y(t′ ) < 1 for any t′ ∈ [t∗ , t+ [,and thusy(t′ ) − y(t+ )ẏ(t+ ) = lim≥ 0.
t′ ↑t+
t′ − t+Meanwhile, because y(t+ ) = 1, we have ẏ(t+ ) = −b(1 − x(t+ )). This implies
that x(t+ ) ≥ 1, which contradicts what we proved in above arguments. This
completes the proof of lemma 1. ■y

(1, 1)

1ẋ = 0(x∗ , y ∗ )ẏ = 0

0

1x

Figure 1: Phase diagram.
L )−(u−pH )]Hereafter, we assume that b > (pH +D)[(d−p. Figure 1 exhibits
d−pL
the phase diagram for equation (1). It is easy to show that the ẋ = 0 curveintersects the ẏ = 0 curve at only one point in ]0, 1[2 , say, (x∗ , y ∗ ).

Lemma 2. The steady state (x∗ , y ∗ ) is asymptotically stable.

Proof. It suffices to show that for every eigenvalue λ of the Jacobian matrixof the right-hand side of (2), its real part is negative.5 Since ẋ = 0 and ẏ = 0
at (x∗ , y ∗ ), its Jacobian matrix at (x∗ , y ∗ ) is)(0−x∗ (1 − x∗ )(d − pL )y ∗ [b − (1 − y ∗ )pL ] −y ∗ ((1 − x∗ )pL + D)
and thus, its characteristic equation isλ2 + y ∗ ((1 − x∗ )pL + D)λ + x∗ (1 − x∗ )y ∗ (d − pL )[b − (1 − y ∗ )pL ] = 0.
Let λ1 , λ2 be the solution of this equation. Then,λ1 + λ2 = −y ∗ ((1 − x∗ )pL + D) < 0,λ1 λ2 = x∗ (1 − x∗ )y ∗ (d − pL )[b − (1 − y ∗ )pL ].
Note that, if λ1 is not a real number, then λ̄1 = λ2 and λ1 + λ2 < 0, and thus
the real part of λ1 , λ2 must be less than 0. Therefore, we can assume that
λ1 , λ2 ∈ R without loss of generality. Then,b − (1 − y ∗ )pL >D[(d − pL ) − (u − pH )]> 0,
d − pL

implying λ1 , λ2 < 0. This completes the proof of lemma 2. ■Finally, we should check that the payoff of player 2 in (x∗ , y ∗ ) exceeds(pH −pL )(d−pL )pL if D < (d−pand b is sufficiently high. To show this, it sufficesL )−(u−pH )to show that if b → ∞, then the payoff of player 2 in (x∗ , y ∗ ) exceeds pL .
H
Actually, if b → ∞, then (x∗ , y ∗ ) → (1, u−p).6 At this point, the payoff ofd−pL
player 2 is(d − pL ) − (u − pH )D > pL .
pH −d − pL
Therefore, for any sufficiently high b, the long run profit of player 2 exceeds
the short run profit pL . This completes the proof. ■Reference[1] Hofbauer, J. and Sigmund, K. (1998) Evolutionary Games and Populational Dynamics, Cambridge University Press, Cambridge.
56

See Pontryagin (1962) for more detailed arguments.
Use l’Hospital’s rule.

[2] Pontryagin, L. S. (1962) Ordinary Differential Equations, Addison-Wesley,
Massachusetts.
[3] Schuster, P. and Sigmund, K. (1983) “Replicator Dynamics”, Journal of
Theoretical Biology 100, 533-538.
[4] Selten, R. (1980) “A Note on Evolutional Stable Strategies in Asymmetrical Animal Conflict”, Journal of Theoretical Biology 84, 93-101.
[5] Taylor, P. D. and Jonker, L. B. (1978) “Evolutionarily Stable Strategies
and Game Dynamics”, Mathematical Biosciences 40, 145-156.
[6] Weibull, J. (1997) Evolutionary Game Theory, MIT Press, Massachusetts.