数理解析研究所講究録 1443 巻 2005 年 164-170

164

A Question on the Usefulness ofCopyright Protection
Yuhki Hosoya
Graduate School of Economics, Keio University.

*Abstract
This paper treats the problem of copyright protection. Many works
state that strong copyright protection increases producer’s surplus anddecreases consumer’s surplus, but we argue that it is doubtful We use
a repeated game to analyze this problem, and show that weakening
copyright protection may improve economic welfare in the sense ofPareto.

1Introduction

The problem finding the optimal level of copyright protection became a serxous economic problem. This problem started since several years ago accompanying with the problem of digital copy, especially illegal copy of music
files.
To understand this problem, we need to consider why copyright is needed.
Firstly, suppose no copyright exists. The cost of copying is usually muchcheeper than that of creating, so we can assume this cost is zero. If theoriginal author sells his works, other firms can copy and sell it with costzero. Because of competition, the price tends to zero, so the original author
suffers a big loss. Therefore, he doesn’t try to sell his works. Next, suppose
copyright exists and is so strong that no firms can copy and sell. Then theoriginal author can gain positive profit. It is not good for economy, however,
because the author gains monopolistic profit.
Therefore, too high or too low level of copyright protection is undesirable.
So we want to know the optimal level of copyright protection. Many works
consider that high level copyright protection increases producer’s surplus anddecreases consumer’s surplus1 .
-mail address; ukki@jcom.home.ne.jp
xFor example, see Landes and Posner(1989)$*\mathrm{E}$165

The aim of this paper is to submit a theoretical counter example on thisstatement. Some firms allow consumer to try their software as long as helikes. Other firms allow consumer to use their software freely, and dependon the donation from users2. The fact these softwares exist and manage asusual tells us an important point. In these examples, author abandons theright to receive money, but his profit is not equal to zero,
Remember our explanation why copyright is needed. We considered theprice competition between the author and the copy firms. In usual pricecompetition it is assumed if the price of one firm is not the minimum price,then the sales of his products must be zero, because consumers can buy
it from other firms whose price is lower than him. A principle behind thisassumption is that; consumers always prefer the cheapest pa ment. However,the case in the previous paragraph is a counter example.
Existence of this counter example raises a question. Examples show thateven if no copyright protection exists, the author can gain positive profit. Can
no copyright protection improve economic welfare, in the sense of Pareto?
We analyze this problem in the framework of repeated game, and answer
this question positively. One example is the case in which consumers’ heterogeneity is very large. It is treated in section 2. The other example isthe case in which an uncertainty about the value of works exists, treated insection 3.

2

The case under large heterogeneity$=\{0$ , ,..,
Consider a repeated game with one supplier and consumers.
below. Inproceeds
means a player set, and player 0 denotes supplier. Gamethe beginning of each period , supplier chooses “make’7 (M) or “halt” (B), If
supplier chooses $H$ , period finishes and every player gains payoff 0. If not,. Consumers choose “buy” (B) ,fromsupplier determines a price“don’t use” (B), or “copy” (C) simultaniously. The payoff of consumer at$n\}$$\parallel$

$n$

$t$

$t$

$[0, \infty)^{3}$$p_{t}$$\mathrm{i}$1sif his action isif his action is0 otherwise$u_{i}-p_{t}$$u_{i}-Xp_{\mathrm{f}}$

$t$

$B$
$C$

where the term $X$ denotes the level of copyright protection. We consider
only two different types of $X$ . One is the case $X=0$ , called no copyright$2\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{t}$paxformance is another example in which the profit of supplier depends on thedonation.
don’t assume that supplier determines
cannot become well-defined$3\mathrm{W}\mathrm{e}$$p_{t}$

from$\mathrm{R}$, because total payoff function

16G

protection case. Another is the case $X=1$ , called full copyright protection
case. If consumers choose $B$ and consumers choose $C$ , the payoff ofsupplier at period is $ktpt+Xl_{t}p_{t}-c$ . The term denotes cost to make hisworks. The total payoff of each player is total sum of payoff with discountrate , that is, if player gains payoff in period , his total payoff is$k_{t}$$l_{t}$$t$$\delta$$c$$\mathrm{i}$$t$$x_{i}^{t}$$\sum_{t=1}^{\infty}\delta^{t-1}x_{i}^{t}$.

The situation this game expresses is here. Supplier can make his works
or not. If he chooses to make it, he must pay a fixed cost . Suppose that
supplier chooses to make his works. If consumer buys and consumes it, hegains the payoff and loses dollars. In no copyright protection case(thatis, $X=0$ ), he can copy freely and consume this works. In full copyrightprotection case(X $=1$ ), he cannot copy and must buy works if he wants toconsume it. Of course, he can ignore supplier’s works.
We can find a general result in the case of no copyright protection.
$c$$\mathrm{i}$$pf$$u_{i}$

theorem 2.1 Suppose $X=0$ and $U=u_{1}+\ldots+u_{n}>c$ . For every
such that, , and $v_{0}+\ldots+v_{n}=U-c$ , if 1– isfor every
sufficiently small,, then there exists a Nash equilibrium in which each player
gains payoff.
$v\in \mathrm{R}_{++}^{n+1}$$\mathrm{i}=1$$v_{i}\leq u_{i}$$\frac{1}{1-\delta}$$\ldots$$n$$\delta$$v_{i}$

proof We consider the sequence of actions$i(\mathrm{m}\mathrm{o}\mathrm{d} n)$, supplier chooses $a_{0}^{t}=(M,$$(a_{0}^{t}, \ldots, a_{n}^{t})_{t}$such that if$\frac{1-\delta^{n}}{\delta^{i-1}(1-\delta)}(u_{i}-v_{i}))$$t\equiv$, only player$\mathrm{i}$chooses $a_{i}^{t}=B$ , and other players choose $a_{j}^{t}=C$ . This sequence of actions
attains desired payoff. Next, we consider a trigger strategy
suchthat in period , player takes
if there was no action differs from for
any $t<t^{*}$ , andorotherwise. By the method similar tothe proof of folk theorem, it can be shown that if
is sufficiently small,then
is a Nash equilibrium,$(s_{0}, \ldots, s_{n})$$t^{*}$$\mathrm{i}$$H(\mathrm{i}\mathrm{f}\mathrm{i}=0)$$a_{i}^{t^{*}}$$a_{i}^{t}$$N(\mathrm{i}\neq 0)$$1-\delta$$(s_{0}, \ldots, s_{n})$$\blacksquare$

example 1 (first example copyright protection is not effective)
Suppose $n=2$ , $u_{1}=0.9$ , $u_{2}=0.2$ and $c=1$ . At first, we show that there
is no equilibrium in which both consumers can consume the works for every
period under the case $X=1$ . We use reductio ad absurdum. Suppose
not. Every player has the minimax value 0, so that, in equilibrium the payoff
vector must belong to. The payoff of player 2 is$t$$\mathrm{R}_{+}^{3}$$\sum_{t=1}^{\infty}\delta^{t-1}(0.2 -p_{b})\geq 0$.

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So, the payoff of player 0 is$\sum_{t=1}^{\infty}\delta^{t-1}(2p_{t}-1)<0$,

a contradiction.

satisfies
Therefore, in equilibrium the payoff vector $U^{*}=$$(1-\delta)(U_{0}^{*}+U_{1}^{*}+U_{2}^{*})<0.1$ and. So, we can find a vector
such that $v_{0}+v_{1}+v_{2}=0.1$ and $v_{l}>(1-\delta)U_{i}^{*}\geq 0$ . Also, we can check $v_{i}<u_{i}$. Using above theorem, we find an equilibrium under $X=0$ suchfor$(U_{0}^{*}, U_{1}^{*}, U_{2}^{*})$$v\in \mathrm{R}^{3}$$0\leq U_{i}^{*}$$\mathrm{i}=1,2^{4}$$\underline{1}v>U^{*}$. Therefore, every equilibrium underthat the payoff vecter is$X=1$ is Pareto dominated by some equilibrium under $X=0$ .
$1-\delta$$\blacksquare$

Next proposition suggests that above example depends on the factlargely differs from .

$u_{1}$$u_{2}$

proposition 2.1 Suppose u $=u_{1}=u_{2}= \ldots=u_{n}>\frac{c}{n}$ . Then, there is anequilibrium such that all consumers can consume the works even if X $=1$ .
proof the couple of actions $a=$ $((M, u)$ , , ..., ) is a Nash equilibriumof component game. So the couple of strategy such that every player takes
action regardless of history is a Nash equilibrium of repeated game.
$B$

$B$

$\blacksquare$$a$

Next proposition is the opposite of above.

proposition 2.2 Suppose U $=u_{1}+\ldots+u_{n}<c$. Then supplier never chooses
M in equilibrium even if X $=0$ .
proof Since 0 is the minimax point, the equilibrium payoff vector $U^{*}=$. If supplier chooses $M$ in period then the, ...,
must belong tosum of payoff in period must be less than 0. Otherwise, the sum of payoff
equal to 0. So if once supplier chooses $M$ , $U_{0}^{*}+\ldots+U_{n}^{*}<0$ , a contradiction.
$\mathrm{R}_{+}^{n+1}$$U_{n}^{*})$$(U_{0}^{*}$$t_{\}}$$t$$\blacksquare$3

The case under some uncertainty

The last proposition of section 2 seems to be obvious. However, if there issome uncertainty about the value of works, we can find a counter exampleof this proposition. In this section, we consider this example.
$4\mathrm{F}\mathrm{o}\mathrm{r}$example, $v_{0}+v_{2}\geq 0$ means $v_{1}\leq 0.1$ $<u_{1}$ .

188The model considered in this section is similar to in previous section,
except if supplier chooses to make his works in period , the value of works$x^{t}=$, . . .,determined randomly. Supplier cannot know
until theend of period ?. Consumer can know
if he consumes works. In full
copyright protection case, consumer cannot consume it except he Paysdollers. Contrastively, in no copyright protection case, consumer can copy
and consume it freely, and determines to buy or not to buy depending on .
The payoff of supplier is the same as previous section. The payoff ofconsumer is rather changing. If he doesn’t choose $N$ , it is expressed by, where denotes the value of works andfunction
the payment.
If he chooses $N$ , then his payoff is .
$t$$(x_{1}^{t}$$x^{t}$$x_{n}^{t})_{t}$$\mathrm{i}$$x_{i}^{t}$$p$$x_{i}^{t}$$u_{i}(x,p^{*})$$x$$p^{*}$$0^{5}$

example 2 (second example copyright protection is not effective)
Suppose $n=1$ , $c=0.08$ , $u(x,p^{*})=-e^{-10(x-p^{*})}+0.17$ and is independentof and has an uniform distribution on.
Firstly, we consider the case of full copyright protection. Suppose
denotes the sum of expected payoffs at period . If supplier chooses $H$ atperiod , then $W_{t}=0$ . If supplier chooses $M$ and consumer chooses $N$ , then$W_{t}=-0.08$ . Otherwise,$x^{t}$$[0, 1]^{6}$$t$$W_{t}$$t$$t$$W_{t}=p-0.08$ $-E[e^{-10x+10p}+0.17]$ $=p+ \frac{e^{10p}(e^{-10}-1)}{10}+0.09$

which depends on . To maximize $W$ , it increases to $\frac{-\log(1-e^{-10})}{10}-0.01<$$p$

where $p= \frac{-\log(1-e^{-10})}{10}$ . So
if and only if supplier chooses $H$ atperiod . Recall that both players have the minimax value 0. Therefore, inequilibrium payoffs must be larger than 0. It means supplier must not choose$M$ in equilibrium.
Next, we treat the case of no copyright protection. Consider a pair ofactions $a=(M, \frac{1}{10})$ and. The expected
payoff of supplier is $0.01>0$ . The expected payoff of consumer is$0$$W_{t}\geq 0$$t$$b=(B( \mathrm{i}\mathrm{f}x\geq\frac{1}{10}), C(\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}))^{7}$.

$\frac{1}{10}(e^{-1}-1)+\frac{e}{10}(e^{-10}-e^{-1})+0.17=\frac{1}{10e}+\frac{1}{10e^{9}}-0.03>0$

Because of the folk theorem, there exists a Nash equilibrium such that players
choose action $(a, b)$ for every period . The payoff of both player in such
equilibrium must be larger than 0.
$t$that wedon’tassume that $u_{i}(0,0)=0$ .
the number of supplier is truely one, the assumption
is independent of is sostrong that it is not practical. Therefore, we explain the supplier includes infinitely many
people in fact. One supplier chooses to make or not and retires at the end of period.
can explain the action , “use as a trial and buy if he likes it”$\overline{\mathrm{s}_{\mathrm{N}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}}}$$6\mathrm{I}\mathrm{f}$$x^{t}$$7\mathrm{W}\mathrm{e}$

$b$

$t$

169

In this example, consumer is risk-averse and supplier is risk-neutral. Consumer can impose his risk on supplier in no copyright protection case, but
it cannot in full copyright protection case. So the supremum of the sum of
expected payoff in no copyright case may become larger than in full copyright case. That is the reason why weakened copyright protection improves

economic welfare.

$\blacksquare$This example shows that no copyright protection may be much better
than full copyright protection under uncertainty. In example 1, every equilibrium payoff on full copyright protection is dominated by some equilibriumpayoff on no copyright protection. On the other hand, in example 2 full
copyright protection interrupts supplier makes his works, so that, every equilibrium payoff on full copyright protection must be equal to 0. Therefore,
every equilibrium payoff on full copyright protection is dominated by every
equilibrium payoff on no copyright protection.

4Conclusion

Example 1 treats the case in which consumers’ heterogeneity is very large,
and shows that weak copyright protection may improve economic welfare.
Example 2 deals with the case in which there is some uncertainty on valueof works, and shows that strong copyright protection may disable supplierfrom making his works.
Both examples reveal that strong copyright protection is often not effective 1n the sense of Pareto. So we should be careful to strengthen the levelof copyright protection.

References[1] Anderson, S. P. and V. A. Ginsburgh(1994) “Price Discrimination via
Second-hand Markets”, European Economic Review, Vol. 38, pp. 23-44.
[2] Arrow, K. J.(1962) “Economic Welfare and the Allocation of Resources
for Invention” , in R. R. Nelson ed., The Rate and Direction of Inventive
Activity, Princeton University Press, pp. 609-626.
[3] Landes, W. M. and R. A. Posner(1989) “An Economic Analysis of Copyright Law” , Journal of Legal Studies, Vol. 18, pp. 325-363

170[4] Shy, 0.(2000) “The Economics of Copy Protection in Software and Other
Medi\"a, in B. Kahin and H. R. Varian eds., Internet Publishing andBeyond, The MIT Press, pp.97-113