Hohjo, Hitoshi

Teraoka, Yoshinobu

Kyoto University Research Information Repository

TitleN-Person Game of Timing on Sale -Silent Case(TheDevelopment of Information and Decision Processes)Author(s)Teraoka, Yoshinobu; Hohjo, HitoshiCitation数理解析研究所講究録 (2006), 1504: 105-109Issue Date2006-07URL http://hdl.handle.net/2433/58495RightType Departmental Bulletin PaperTextversionpublisherKyoto University$\mathrm{N}$-Person Game of Timing on Sale-Silent Case(Yoshinobu kraoka)(Hitoshi Hohjo)Department ofMathematics and Information SciencesGraduate School of ScienceOsaka Prefecture University$\mathrm{A}\mathrm{b}\mathrm{s}\mathrm{U}\mathrm{a}\mathrm{c}\mathrm{t}$We consider a class oftiming game which is suggested from a timing problem for putting somekind of farm products on the market. $\mathrm{N}$ players, Player 1, 2, $\cdots,$ $\mathrm{n}$, take possession of the rightto put some kind of farm products at any time in $[0,1]$ . The price of them increases with timeso long as none of the $\mathrm{n}$ players do not sell them, however, if one of the $\mathrm{n}$ players puts his farmproducts on the market the price s discontinuously and then increase with time. Suchdiscontinuous falls in the price of farm product arise successively until $\mathrm{n}$ players complete to$\mathrm{s}\mathrm{e}\mathrm{U}$. All players have to put their farm product within the unit interval 10, 11. In such asituation, each player wishes to delay action as late as possible, but does not wish to delay solate that his opponents can put earlier. We assume that all players do not learn when norwhether their opponents have put their arm products on the market. Each of the $\mathrm{n}$ playershas to decide his action time. This model yields us a $\mathrm{c}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{a}\dot{\mathrm{i}}$ class of $\mathrm{n}\cdot \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{n}\cdot \mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}$ suminfinite game.1. Introduction We consider a class of game which is suggested from correlativephenomena between the price and supply in a market on farm products. $\mathrm{N}$ players,Player 1, $\cdots,$ $\mathrm{n}$ take possession of the right to put some kind of firm products onthe market with same ratio. We call such kind of products product $\mathrm{F}$ in this paper.We can harvest product $\mathrm{F}$ at a specific season every year periodically. Each of the$\mathrm{n}$ players wants to decide the optimal time to put his product $\mathrm{F}$ on the marketuntil the next harvest season. We consider one time period where the harvest timein each year is the beginning and the next harvest time is the end. The price ofproduct $\mathrm{F}$ increases with time so long as none of the players don’t put on themarket and keep their own products. But, when one of the players put his product$\mathrm{F}$ on the market, the price of product $\mathrm{F}$ possessed by the other n-l players falls1504 2006 105-109 105discontinuously and then increases with time continuously. Such discontinuousfalls in the price on product $\mathrm{F}$ possessed by the other players arise successivelywhenever one of the rest players put his product $\mathrm{F}$, until all on the $\mathrm{n}$ playerscomplete to sell. In such a situation, each player wishes to delay action as late aspossible to increase the price of his product $\mathrm{F}$, but does not wish to delay so latethat the price falls discontinuously several times without knowing the currentprice. Each player has to decide his optimal action time considering the currentprice and his opponents’ action times, each other.As well as the usual games of timing [1, 21, there are two $\mathrm{k}_{\dot{\mathrm{i}}}\mathrm{d}$ of informationavailable to the players. If a player is informed of his opponents’ action times assoon as one of his opponents put his product $\mathrm{F}$ on the market, we say they are in anoisy version. If neither player learns when or whether each of the $\mathrm{n}\cdot 1$ players hasput his product $\mathrm{F}$ on the market, we say they are in a silent version. We considerthe case where $\mathrm{n}\cdot \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}\mathrm{s}$ are in a silent version in this paper.This model yields us a certain class of $\mathrm{n}\cdot \mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{o}\mathrm{n}\cdot \mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}$ sum infinite gameson the hyper $\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\cdot \mathrm{c}\mathrm{u}\mathrm{b}\mathrm{e}$ . The equilibrium strategies express the proper balancebetween the desire for delay and the danger of delaying. Related to our model, wecan point out three works [3, 4, 5].2. Notations and Assumptions Since we consider one period game, we expressthe period as the unit interval $[0,1]$ . Throughout of this paper, we use thefollowing notations:$v(t)$ is the price of product $\mathrm{F}$ at time $t\in[0,1]$ , when any of the $\mathrm{n}$-players doesn’tput his product F. We assume that $v(t)\mathrm{i}\mathrm{s}$ differentiable and $v’(t)>0$ for $t\in(0,1)$ .It is natural to assume $0<v(\mathrm{O})<\infty$ .$r$ is the discount factor after one of the $\mathrm{n}$-players has already put his product $\mathrm{F}$on the market and is assumed $0<r<1$ . That is, if one of the $\mathrm{n}\cdot \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}\mathrm{s}$ sells hisproduct $\mathrm{F}$ at time $t\in[0,1]$ , the price of product $\mathrm{F}$ possessed by other $\mathrm{n}\cdot 1$ playersfalls down from $v(t)$ to $rv(t)$ immediately. Hence if $\mathrm{k}$ players have already puttheir product $\mathrm{F}$ in the market before time $t\in[0,1]$ , the price of it possessed by othern-k players decreases to $r^{k}v(t)$ at time $t\in[0,1]$ .If $\mathrm{k}$ players put their product $\mathrm{F}$ at a same time $t\in[0,1]$ , each of the $\mathrm{k}$ players canget the current price $\hat{v}(t)$ , and then the price of product $\mathrm{F}$ decreases to $r^{k}\hat{v}(t)$ .We now denote the expectations for the real valued function $M_{j}(x_{1}$ , —, $x_{n})$over [$0$ , llx $\mathrm{x}[0,1]$ when player $i$ uses $cdfF_{i}(x_{l})$ over $[0,1]$ as his mixed106strategy as follows:$M_{i}(F_{1},F_{2},---,F_{n})= \int--- \mathrm{J}M_{i}(x_{1},x_{2},---,x_{n})dF_{1}(x_{1})dF_{2}(x_{2})--- dF_{n}(x_{n})$and$M_{i}(x_{1},F_{2},---,F_{n})= \int M_{i}(x_{1},x_{2}.---,x_{n})dF_{2}(x_{2})--- dF_{n}(x_{n})$ .3. Formulations and Analysis Since each of the $\mathrm{n}\cdot \mathrm{p}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}\mathrm{s}$ can’t observe theother $\mathrm{n}\cdot 1$ players’ action time, he decide his putting time in advance and then hecan find the value of his product $\mathrm{F}$ after putting on the market. Hence weestablish the pure strategy for Player $i$ as $x_{i}\in[0,1],$ $i=1,2,—,n$ Then theexpected payoffkernel to Player 1 $M_{1}(x_{1},x_{2},---,x_{n})$ is given by$\mathfrak{l}_{l}^{\mathrm{J}}.\prime\prime$$\mathrm{v}(\mathrm{x}_{\mathrm{i}})$ , $0\leqq \mathrm{X}\iota\leqq \mathrm{y}_{(1\rangle}$$.\mathit{1}^{t}|_{1?_{\iota_{\dot{l}}}}\mathrm{r}^{2}\mathrm{v}(\mathrm{x}_{1})\mathrm{r}\mathrm{v}(\mathrm{x}_{1})$$\mathrm{y}_{(1)}<\mathrm{x}_{1}\leqq \mathrm{y}_{(2)}$$\mathrm{y}_{(2)}<\mathrm{x}_{1}\leqq \mathrm{y}_{(3)}$$\mathrm{M}_{1}$$(\mathrm{x}_{1},\mathrm{x}_{2},-.. ’ \mathrm{x}_{n})=$$\backslash _{1}’$-. $—-$ -.....-...–..... $—-$ (1)$.\}tfd$’$\mathrm{r}^{\iota}\mathrm{v}(\mathrm{x}_{1})$$\mathrm{y}_{\langle k)}<\mathrm{x}_{1}\leqq \mathrm{y}_{(k+1)}$$\mathrm{i}\mathrm{I}_{i\{}$..................-.........$:\backslash ^{\mathrm{r}^{n- 1}\mathrm{v}(\mathrm{x}_{1})}$ $\mathrm{y}_{n- 1}<\mathrm{x}_{1}\leqq 1$ ,where $y_{(l\rangle}$ denotes the $i$ th smallest of $x_{2},---,x_{n-1},$ $i=1,—,n-1$ . Hence, wesuppose that $\mathrm{y}_{(1)}\leqq \mathrm{y}_{(2)}\leqq\ldots\leqq \mathrm{y}_{(n- 1)}$ .Observing the above payoff kernel, we can suppose that all of the $\mathrm{n}$ players usesame mixed strategy (cdfon $[0,1]$) $F(x)$ and also suppose that $F(x)$ consists ofits pdf $f(x)>0$ over an interval $(a, 1)\subset[0,1]$ by considering the result of twoperson $\mathrm{g}\mathrm{a}\mathrm{m}\mathrm{e}[5]$ . Since the expected payoff to Player 1 $M_{1}(x,F,---,F)$ for any$x\in(a, 1]$ is given by$\mathrm{M}_{1}(_{\mathrm{X},\mathrm{F},\mathrm{F},-,\mathrm{F})=\mathrm{v}\mathrm{t}\mathrm{x})_{[\sum_{k\cdot 0}^{n- 1}}}$-l’ (2)107we have$\mathrm{M}_{1}(\mathrm{x},$ $\mathrm{F},$ $\mathrm{F},$$\cdot$ -. , $\mathrm{F}\rangle$ $=v(x)$ , $0\leqq \mathrm{x}<\mathrm{a}$$=v(x)[1-(1-r)F(x)]^{n- 1}$ , $\mathrm{a}\leqq \mathrm{x}\leqq 1$ . (8)Putting$M_{1}(x,F,F,---,F)=con\mathrm{s}t$ for $x\in(\mathrm{a},$ $1\rangle$ ,we get$v(x)[1-(1-r)F(x)]=(n-1)(1-r)f(x)v(x)$, $a<x<1$ ,which yields$F(x)=\{1/(1-r)\}[1-\{c/v(x)\}^{1/(n=1)}]$ , $a<x<1$ , (4)where $c$ is $an$ integration constant. The boundary conditions$F(a)=0$ and $F(1)=1$give us$c=r^{n- 1}v(1)$ $j$ $v(a)=r^{n- 1}v(a)$ .However, the above conditions are satisfied even when $v(0)\leqq r^{n- \mathrm{l}}v(1)$ .So we consider the case where $v(0)\leqq r^{n-1}v(1)$ first. For this case, thereexists the unique root $a$ which satisfies the equation $v(a)=r^{n-1}v(1)$ in theinterval $[0,1]$ . And then we have the following relations:$M_{1}(x,F,F,---,F)=v(x)<v(a)=r^{n-1}v(1)$ , $0\leqq x<a^{0}$$=v(a)=r^{n-1}v(1)$ , $a^{0}\leqq x\leqq 1$ (5)Hence we get Theorem 1.Theorem 1. Assume that $v(0)\leqq r^{n-1}v(1)$ . Let $a^{0}$ be the unique root of equationof $v(a)=r^{n-1}v(1)$ in the interval [ $0,$ 11. And consider the following mixedstrategy:$F^{0}(x)=0$ , $0\leqq x<a^{0}$$=\{1/(1-r)\}[1-\{v(a)/v(x)\}^{1/(n-1)}]$ , $a^{0}\leqq x\leqq 1$ .Then n-tuple of mixed strategies $(F^{0},F^{0},---,F^{0})$ is a Nash equilibrium of ourn-person non-zero sum silent game. And the corresponding equilibrium value $v_{i}$to Player $i$ is given as$v_{i}=M_{i}(F^{0},F^{0},---,F^{0})=r^{n-1}v(1)$ , $i=1,—,n$ .108Now we consider the case where $v(0)>r^{n- 1}v(1)$ . Since we assumed that each of$\mathrm{k}$ players can get the current price $\hat{v}(t)$ when $\mathrm{k}$ players put their product $\mathrm{F}$ at asame time $t\in[0,1]$ , we have$M_{1}(x,\mathrm{O},---,\mathrm{O})=r^{n-1}v(x)\leq rv(1)<v(\mathrm{O})$, $0<x\leq 1$ (6)and also have$M_{1}(0,x_{2},---,x_{n})=v(0)>r^{n-1}v(1)$ , for any $x_{2},---,\chi_{n}$ in $[0, 1]$ . (7)Thus we get Theorem 2.Theorem 2. Assume $v(\mathrm{O})>r^{n-1}v(\mathrm{O})$ . Then $\mathrm{n}\cdot \mathrm{t}\mathrm{u}\mathrm{p}\mathrm{l}\mathrm{e}$ (0,—,0) is a Nashequilibrium point of this non-zero sum silent game. The correspondingequilibrium value $v_{l}$, to Player $i$ is given as$v_{i}=M_{i}(0,---,0)=v(0)$, $i=1,—,n$ .If $\mathrm{n}$ is greater than some value the inequality $v(\mathrm{O})>r^{n-1}v(\mathrm{O})$ holds. It meansthat when the number of participants of such a game exceeds some level, each ofthem wants to $\mathrm{s}\mathrm{e}\mathrm{U}$ his product $\mathrm{F}$ at the first time of the game.References1. M.Dresher, Games of Strategy : $Theo\eta^{r}$ and Applications, $\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}^{-}\mathrm{H}\mathrm{a}\mathrm{l}1$,Englewood Cliffs. New Jersey, 1554.2. S.Karlin, Mathematical Method and Theory in Games, Programming andEconimics Vol.2, $\mathrm{A}\mathrm{d}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{n}^{-}\mathrm{W}\mathrm{e}\mathrm{s}\mathrm{l}\mathrm{e}\mathrm{y}$, Massachusetts, 1959.3. Y.Tbraoka and Y.Yamada, Games of production development in manufacturing,Lecture Note in Economics and Mathematical Systems 445, Stochastic$Modelb\dot{n}gin$ Innovative $Manuf_{\mathit{9}Ctu\mathit{1}\dot{\mathrm{Z}}I\mathit{1}}g$, Springer, Berlin, 1997, 58.67.4. Y.?braoka and H.Hohjo, N.person games on territory, Game Theory andApplxationsVol.V, Nova Science Publishers, Inc., New York, 2000, 134.141.5. Y.braoka and H.Hohjo, Two person games of timing on sale, Proceedings ofInternational Workshop on Recent Advances in Stochastic OperationsResearch, Nanzan University, Nagoya, 2005, 281-189.109

N-Person Game of Timing on Sale -Silent Case(The Development of Information and Decision Processes)

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N-Person Game of Timing on Sale -Silent Case(The Development of Information and Decision Processes)

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