自発的繰り返し囚人のジレンマにおける確率的受け入れと漸次協力の効果について On Stochastic Acceptance and Gradual Cooperation in Voluntarily Repeated Prisoner's Dilemma with No Information

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1 Powered by TCPDF ( Title Sub Title Author Publisher 自発的繰り返し囚人のジレンマにおける確率的受け入れと漸次協力の効果について On stochastic acceptance and gradual cooperation in voluntarily repeated prisoner's dilemma with no information flow グレーヴァ, 香子 (Fujiwara-Greve, Takako) 慶應義塾経済学会 Publication year 2008 Jtitle 三田学会雑誌 (Keio journal of economics). Vol.101, No.3 ( ),p.465(69)- 489(93) Abstract 近年, 自発的繰り返しゲームの研究が進んできた自発的繰り返し囚人のジレンマにおいて協力を達成するには, 常に協力したときの利得より生涯利得を低くしなくてはならないその方法として, 出会ってもパートナーとして受け入れない確率を導入する ( 確率的受け入れ ) か, プレイはするが最初の何回かは協力しない ( 漸次協力 ) 戦略が均衡として示されてきた本稿ではこれらの均衡の利得を比較することで, どちらの戦略が相対的に望ましいかを考える Lately, progress has been made in the research of voluntarily repeated games. To achieve cooperation in a voluntarily repeated prisoner's dilemma, the lifetime payoff must be set lower than the total payoff from repeated cooperation. Two kinds of equilibria are proposed to generate this structure: stochastic acceptance equilibria where players accept a newly matched partner only with some probability less than one to start the game, and gradual cooperation equilibria where players accept each other for sure but do not cooperate in some initial periods. In this study, I compare the equilibrium payoffs of these equilibria to determine which strategy is better. Notes 小特集 : 経済の数理 : 非線形動学と経済の変動を中心に Genre Journal Article URL

2 自発的繰り返し囚人のジレンマにおける確率的受け入れと漸次協力の効果について On Stochastic Acceptance and Gradual Cooperation in Voluntarily Repeated Prisoner's Dilemma with No Information Flow グレーヴァ香子 (Takako Fujiwara-Greve) 近年, 自発的繰り返しゲームの研究が進んできた自発的繰り返し囚人のジレンマにおいて協力を達成するには, 常に協力したときの利得より生涯利得を低くしなくてはならないその方法として, 出会ってもパートナーとして受け入れない確率を導入する ( 確率的受け入れ ) か, プレイはするが最初の何回かは協力しない ( 漸次協力 ) 戦略が均衡として示されてきた本稿ではこれらの均衡の利得を比較することで, どちらの戦略が相対的に望ましいかを考える Abstract Lately, progress has been made in the research of voluntarily repeated games. To achieve cooperation in a voluntarily repeated prisoner s dilemma, the lifetime payoff must be set lower than the total payoff from repeated cooperation. Two kinds of equilibria are proposed to generate this structure: stochastic acceptance equilibria where players accept a newly matched partner only with some probability less than one to start the game, and gradual cooperation equilibria where players accept each other for sure but do not cooperate in some initial periods. In this study, I compare the equilibrium payoffs of these equilibria to determine which strategy is better.

3 JEL classification C Aumann (1981) Fudenberg and Maskin (1986) Matsushima (1990) Kandori (1992) Ellison (1994)

4 1 (Voluntarily Repeated Games) Fujiwara-Greve and Okuno-Fujiwara (2008) Voluntarily Separable Repeated Prisoner s Dilemma 2 Datta (1996) Ghosh and Ray (1996) Kranton (1996)

5 (Fudenberg and Maskin (1986)) 1 (Kandori (1992) Ellison (1994)) 3 2 Fujiwara- Greve (2002) Fujiwara-Greve and Okuno-Fujiwara (2008) Fujiwara-Greve and Okuno-Fujiwara (2008) 4 (trust building)

6 2 Fujiwara-Greve (2002) Fujiwara-Greve and Okuno-Fujiwara (2008) 2 2 [0, 1] τ =1, 2,... δ (0, 1) τ =1 τ =2, 3,... p u a Accept r Reject a 1 r u 1 2 C D g>c>d>l

7 1 P1\ P2 C D C c, c l, g D g, l d, d 1 D C D 2c >g+ l C c >u u >d m Maintain e End 1 q m t =1, 2,... 1 H 1 = { } t =1 t =2, 3,... H t =({C, D} {C, D}) t 1 t 5 s =(f A, {ft G } t=1, {ft C } t=1)

8 1 [] f A : H 1 {a, r} a r [ ] t =1, 2,... ft G : H t {C, D} [ ] t =1, 2,... ft C : H t ({C, D} {C, D}) {m, e} τ =1, 2,... t s s τ 6 1 S S S Σ = (S S) s

9 s s s U(σ) =E δ τ 1 u(τ; σ) τ=1 σ u(τ; σ) σ τ σ (sequential equilibrium) 7 (belief system) X 8 P X P I (belief system) µ : X [0, 1] 1 I P x I µ(x) =1 σ µ (σ, µ) (sequential equilibrium) 2 [ ] I µ σ σ [ ] σ {σ k } k=1,2,... k =1, 2,... σ k σ k µ k : X [0, 1] µ 7 Kreps and Wilson (1982) Abreu et al. (1986), Kandori and Matsushima (1998) 8 (1997)

10 1 2 s σ s σ s s s s 2 2 Okuno-Fujiwara and Postelwaite (1995) A B s r D σ s

11 α s(α) α a (C, C) C D (C, C) m s(α) U V U V U = p{αv +(1 α)(u + δu)} +(1 p)(u + δu), V = c + δ{qv +(1 q)u} p α V u 9 δu 1 p u δu 2 s(α) (C, C) C c q V 1 q U 12 U = V = αpc +(1 δq)(1 αp)u (1 δ){1 δq(1 αp)}, 3 {1 δ(1 αp)}c + δ(1 q)(1 αp)u (1 δ){1 δq(1 αp)} 4 9 U U

12 3 4 (g, c, d, l) (p, q, δ, α) V (u + δu) = c u 1 δq(1 αp) > 0 5 (1 αp)(c u) V U = 1 δq(1 αp) > p (0, 1] α (0, 1] s(α) g c (g u)δq < 1 7 s(α) 1 s(α) s(α) 1. s(α) r u + δu V 5 2. D s(α) V D g g + δu V (g + δu) = 0 δq(1 αp) = g c g u 8 3. D e (C, C) e U s(α) V

13 s(α) 8 α 5 { 1 g c (g u)δq } 1 p α>0 p (0, 1] g c (g u)δq < 1 1 (Fujiwara-Greve (2002)) s(α) (p, δ, q, α) (0, 1] (0, 1) (0, 1) (0, 1] δq αp δq 7 10 p p p {( α (p) :=min 1 g c ) 1 } (g u)δq p, 1 p 0 1 α (p) 1 1 g c (g u)δq p α =1 p α (p) < 1 p p =1 g c (g u)δq p 5 p α =1 U(s(1)) = pc +(1 δq)(1 p)u (1 δ){1 δq(1 p)}

14 U s(α (p)) 12 s(1) p p U(s(1)) p = (1 δq)(c u) (1 δ){1 δq(1 p)} 2 > 0 p p p α (p) U(s(α (p))) = δqg (g c) δq(1 δ) 9 U p 5 p 2 g =10,c =6,d =2,l =0,δ =0.8,q =0.7 c 1 δ c δqg (g c) (1 δq)(g c) = > 0 1 δ δq(1 δ) (1 δ)δq p U(s(α (p))) Fujiwara-Greve et al. (2008)

15 3.2 α A (0, 1) α α A 1 α A A A = {α 1,α 2,...,α K } k α K =1 α 1, α K 1 =1 α 2 A 0.5 A 0.5 α 1 α A p α α (p) k =1, 2,...,K p αk α k = α (p αk ) α M A p αm if p 5 p ˆα (p) = α 1 if p <p5 p α1 α k if p αk 1 <p5 p αk for k =2, 3,...,M p 5 p αk ˆα (p) α (p) 3 p αm p 8 α 2 (p αk 1,p αk ] s(α k ) p p = p αk T D C s (T )

16 3 {0.2, 0.5, 0.8} ˆα * (p) α * (p) p 0.8 p 0.5 p 0.2 p* p U(s(0.8)) U(s(0.5)) U(s(0.2)) 12 p p* p 0.8 p 0.5 p a 2. t 5 T D (D, D) m 3. t>t C (C, C) m s (T ) T s (T ) U

17 t V (t) U =(1 p)(u + δu )+pv (0) V (0) = {1+δq + + δ (T 1) q (T 1) }d +(δ T q T + )c +δ(1 q){1+δq + δ 2 q 2 + }U, = 1 δt q T 1 δq d + δt q T δ(1 q) c + 1 δq 1 δq U. U = (1 δq)(1 p)u + p{(1 δt q T )d + δ T q T c} (1 δ){1 δq(1 p)} 10 V (t) t 5 T V (t) ={1+δq + δ 2 q δ (T t 1) q (T t 1) }d + {δ (T t) q (T t) + }c t = T +1 +δ(1 q){1+δq + δ 2 q 2 + }U = 1 δ(t t) (T t) q d + δ(t t) (T t) q δ(1 q) c + 1 δq 1 δq 1 δq U, V (t) =V (T +1)=(1+δq + δ 2 q 2 + )c + δ(1 q)(1 + δq + δ 2 q 2 + )U = 1 δ(1 q) c + 1 δq 1 δq U t 5 T +1 (D, D) V (t) >V (t 1) T +1 (C, C) T +1 V (t) t V (0) (u + δu )= δt q T (c d) (u d), 1 δq(1 p) V (0) U = (1 p){δt q T (c d) (u d)} 1 δq(1 p) V (0) = u + δu V (0) = U (δq) T (c d) = (u d) 11 V (t) >V (t 1) t = 1 V (t) >u+ δu V (t) >U

18 2 T s (T ) 11 (δq) T (c d) 5 (u d)+ 1 δq(1 p) {δq(g u) (g c)} δpq s (T ) V (0) u + δu 11 V (0) >u+ δu 2. t 5 T s (T ) D V (t) C l + δu 11 t = T +1 s (T ) C V (T +1) D g + δu {V (T +1) (g + δu )}(1 δq){1 δq(1 p)} = {1 δq(1 p)}{δq(g u) (g c)} + δpq(u d) δpq(δq) T (c d) t 5 T (D, D) s (T ) V (t +1) U 11 t = T +1 (C, C) s (T ) s (T ) s (T ) p (0, 1) T h(p) :=(u d)+ 12 (δq) T (c d) 5 h(p) 1 δq(1 p) {δq(g u) (g c)} δpq h (1 δq){δq(g u) (g c)} = p δp 2 q

19 h(p) <p s (0) p p c-d (δq)1(c-d) u-d (δq)2(c-d) p s (1) p p 1 s (2) (11) p p 1 7 h p p h(p) u d 4 T =0 T = c d(> u d) p p 5 p c d 5 h(p) 0 <p5 p s (0) s (0) s(1) p 1 (0, 1) δq(c d) =h(p 1) s (1) δq(c d) = u d p 5 p 1 T 11 T T =0, 1,...,T p T (0, 1) (δq) T (c d) =h(p T ) p 5 p T s (T ) p 0 = p h 4 T p T <p T +1 s (T ) s (T +1)p T <p5 p T +1 D s (T ) T T 11 p T

20 U (s (0)) U (s (1)) p* p 1 p T p T < 1 T p U (s (0)) 0 <p5 p 0(= p ) U (s (T )) p T 1 <p5 p T T =1, 2,...,T, U (s (T +1)) p T <pand T +15 T. 5 T p = p T U (s (T )) = = 1 (1 δ){1 δq(1 p)} { +p d +(u d)+ δqg (g c) δq(1 δ) [ (1 δq)(1 p)u 1 δq(1 p) }] {δq(g u) (g c)} δpq 9 p T 1 p T p 11 U (s (T )) p p = (1 δq){(1 δt q T )d + δ T q T c u} (1 δ){1 δq(1 p)} 2 >

21 U(s(0.8)) U(s(1)) =U (s (0)) 14 6 U(s(0.5)) α*(p) U (s (1)) 13 U(s(0.2)) p p* p 0.8 p 0.5 p 1 p p <p s(1) s (0) 0.5 s (1) 3 A δq(c d) >u d δq 5 (g u)(u d)+(c u)(c d) (g u)(c d) 13 α A p α (p,p 1) p (p,p α ) s(α) p

22 s (1) s(α) s (1) α U(s(α)) = U (s (1)) α = α p p = p δq(c d) (u d) =: α(p) δpq(c d)+(c u) α (p) = α(p) p 5 p 1 14 p 1 h(p 1)=(u d)+ 1 δq(1 p 1) {δq(g u) (g c)} = δq(c d) δp 1q δp 1q(u d)+{1 δq(1 p 1)}{δq(g u) (g c)} = δ 2 q 2 p 1(c d) p 1{δq(u d) δ 2 q 2 (c d)+δ 2 q 2 (g u) δq(g d)} = (1 δq){δq(g u) (g c)} p 1 = δq(g u) (g c) δq{(g u) (c d)} B := δq(g u) (g c) α (p) = α(p) B δq(c d) (u d) = p δq(g u) δpq(c d)+(c u) Bδpq(c d)+b(c u) = δpq(g u){δq(c d) (u d)} [ δpq (g u){δq(c d) (u d)} B(c d)] 5 B(c u) [ δpq (g u)δq(c d) (g u)(u d) δq(g u)(c d)+(g c)(c d)] 5 B(c u) [ δpq {g u) (c u)}(c d) (g u)(u d)] 5 B(c u) δpq(c u){(g u) (c d)} 5 B(c u) δq(g u) (g c) p 5 δq{(g u) (c d)} = p α(p )

23 0.8 α*(p) 7 α α α(p) p* p α p 1 p α(p (g u){δq(c d) (u d)} )= (g u){δq(c d) (c d)} +(c u){(g u)+(c d)} 5 1 2, 2(g u){δq(c d) (u d)} 5 (g u){δq(c d) (c d)}+(c u){(g u)+(c d)} δq(g u)(c d) 5 (g u){2(u d) (c d)+(c u)} +(c u)(c d) (g u)(u d)+(c u)(c d) δq 5. (g u)(c d) A 0.5 α α = 0.5 = α(p ) p>p α>α(p) p α α = α (p α ) α α(p 1)=α (p 1) p α <p 1 7 p α p α 5 α (p) s(α) α>α(p) s (1) δq(c d) >u d s (1) p (p,p 1) 3 c 1 c d = g u δ (0, 1) q (0, 1)

24 (g u)(u d)+(c u)(c d) = 1 (g u)(c d) (g u)(u d)+(c u)(c d) = (g u)(c d) (c u)(c d) = (g u)(c d u + d) c d = g u c d = g u 13 1 δq δq(c d) >u d 13 δq (g = 10, c =6,u =2,d =0,δ =0.8, q =0.7) α(p) 0.31 p = p 0.25 p = p 1 α =0.8 α =0.5 s (1) α = Carmichael and MacLeod (1997) ŝ(α, T ) 1. α 2. t 5 T D (D, D)

25 8 12 ĥ(0.5, p ) 10 h(p) 8 ŝ(0.5,0) c-d 6 (δq)(c-d) u-d 4 2 p p* p 0.5 p 1 p 0.5 ŝ(0.5,1) m 3. t>t C (C, C) m ŝ(α, T ) 4 p αp (δq) T (c d) 5 (u d)+ 1 δq(1 αp) {δq(g u) (g c)} =: ĥ(α, p) δαpq 15 h p ĥ α, p α 15T p α =0.5 6 p 1 p ŝ(0.5, 1) p>p (p 5 p )

26 70 p p 1 (D, D) 1 11 δq d 13 p (D, D) Fujiwara- Greve and Okuno-Fujiwara (2008) 11 Ghosh and Ray (1996)

27 Abreu, D., D. Pearce, and E. Stachetti (1986). Optimal Cartel Equilibria with Imperfect Monitoring. Journal of Economic Theory, 39 pp Aumann, R. (1981). Survey of Repeated Games. In Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern, Mannheim. Bibliographisches Institut. Carmichael, L. and B. MacLeod (1997). Gift Giving and the Evolution of Cooperation. International Economic Review, 38 pp Datta, S. (1996). Building Trust. Manuscript. London School of Economics. Ellison, G. (1994). Cooperation in the Prisoner s Dilemma with Anonymous Random Matching. Review of Economic Studies, 61 pp Fudenberg, D. and E. Maskin (1986). The Folk Theorem in Repeated Games with Discounting or with Incomplete Information. Econometrica, 54 pp Fujiwara-Greve, T. (2002). On Voluntary and Repeatable Partnerships under No Information Flow. In Proceedings of the 2002 North American Summer Meetings of the Econometric Society ( Fujiwara-Greve, T. and M. Okuno-Fujiwara (2008). Voluntarily Separable Repeated Prisoner s Dilemma. Forthcoming in the Review of Economic Studies. Fujiwara-Greve, T., M. Okuno-Fujiwara, and N. Suzuki (2008). Voluntarily Separable Repeated Prisoner s Dilemma with Reference Letters. Manuscript, Keio University, University of Tokyo, and Komazawa University. Available at papers.html. Ghosh, P. and D. Ray (1996). Cooperation in Community Interaction without Information Flows. Review of Economic Studies, 63 pp Kandori, M. (1992). Social Norms and Community Enforcement. Review of Economic Studies, 59 pp Kandori, M. and H. Matsushima (1998). Private Observation, Communication and Collusion. Econometrica, 66 pp Kranton, R. (1996). The Formation of Cooperative Relationships. Journal of Law, Economics &Organization, 12 pp Kreps, D. and R. Wilson (1982). Sequential Equilibria. Econometrica, 50 pp Matsushima, H. (1990). Long-Term Partnership in a Repeated Prisoner s Dilemma with Random Matching. Economics Letters, 34 pp Okuno-Fujiwara, M. and A. Postelwaite (1995). Social Norms and Random Matching Games. Games and Economic Behavior, 9 pp

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.~:ril.~, ー.. ~ 諭 Powered by TCPDF (www.tcpdf.org) Title 歴史の裏面 Author 久米, 邦武 Publisher 三田学会 Jtitle 三田学会雑誌 (Keio journal of economics). Vol.3, No.6 (1910. 6),p.705(77)- 715(87) Abstract Genre Journal Article URL http://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=an00234610-19100615