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Tag-based indirect reciprocity

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Masuda and Ohtsuki. Proceedings of the Royal Society B: Biological Sciences, 274, 689-695 (2007).

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Tag-based indirect reciprocity

  1. 1. Tag-based indirect reciprocity by incomplete social information Naoki Masuda1 and Hisashi Ohtsuki2 The University of Tokyo, Japan 2 Harvard University http://www.stat.t.u-tokyo.ac.jp/~masuda 1 Ref: Masuda & Ohtsuki, Proc. R. Soc. B, 274, 689-695 (2007).
  2. 2. Prisoner’s Dilemma Opponent Cooperate Defect Cooperate (3, 3) (0, 5) Defect (5, 0) (1, 1) Self unique Nash equilibrium
  3. 3. A Prisoner’s Dilemma • A donor may donate cost c to benefit the recipient by b (>c). • If each player serves as donor and recipient in different (random) pairings, the game is symmetric PD. recipient donor C (-c, b) D (0, 0) (b > c) C D C (b-c, b-c) (-c, b) D (0, 0) (b, -c)
  4. 4. Origins of altruism • Kin selection • Direct reciprocity – Iterated Prisoner’s dilemma • • • • • Spatial reciprocity Indirect reciprocity Network reciprocity Group selection Others • Is ‘helping similar others’ a viable (stable) strategy?
  5. 5. An affirmative answer by Riolo, Cohen & Axelrod, Nature 2001 • b=1.0, c=0.1 • Player i has – Tag wi ∈ [ 0,1] – Tolerance µi ∈ [ 0,1] • i cooperates with j if w j − wi ≤ µi • Players copy tag and tolerance of successful others. • mutation: – Random allocation of tag – Neutral drift of tolerance • Results of their numerical simulations of evol dynamics: – Donation rate is maintained high (~ 75%). – The mean tolerance level is small (0.01-0.03). – With some sudden changes though.
  6. 6. However, rebuttal by Roberts & Sherratt (Nature 2002) Criticism 1 i was assumed to coopreate if w j − wi ≤ µi & µi ∈ [ 0,1] Criticism 2 Neutral drift & µi ∈ [ 0,1] A player cooperate with birds with exactly the same feather Random walk with reflecting boundary Cooperation is lost if µi ∈ [ 0,1] is replaced by µi ∈ [ − 10 −6 ,1] Positive bias. Why mutation increases generosity?
  7. 7. We establish a viable model of tagbased reciprocity. [ ] µi ∈ − 10 −6 ,1 • Use a kind of • q: prob that μj is public to others • If player i gets to know μj <|wj-wi|, i does not donate even if μi ≥|wj-wi| • q=0 → eventually ALLD (μi <0) • q=1 → eventually ALLC (μi takes max) • No mutation of tags
  8. 8. 2-tag model • Same or different only. tag tolerance wi ∈ [ 0,1] [ ] µi ∈ − 10 ,1 −6 { → wi ∈ w , w → a b } µi ∈ { − 1,0,1} μ phenotype -1 no donate (D) 0 tag user 1 donate (C)
  9. 9. Payoffs of 6 subpopulations tag = a tag = b h: assortativity
  10. 10. Replicator dynamics • Symmetric case – Full theoretical analysis (global analysis) • Asymmetric case 6 vars, 4 dim note: no tag evolution – Best-response theory (local analysis only) – Numerical simulations
  11. 11. Symmetric case Small q μ phenotype -1 no donate (D) 0 tag user 1 donate (C) Intermediate q Large q c (1 − q ) A≡ < 1 ⇒ bq > c ( b − c) q is the condition for tag users to emerge.
  12. 12. With assortativity h b = 1, c = 0.3 q = 0.5, h = 0 q = 0.5, h = 0.8
  13. 13. Asymmetric case (best response) A= p >A b 1 p1b ≤ A A − (1 − ( t + h − ht ) ) p1b X= . t + h − ht μ Among 9 pure strategies, only (μa,μb)=(-1,-1), (1,0), (0,-1), (0,0), and (1,1) are viable. c (1 − q ) , ( b − c) q phenotype -1 no donate (D) 0 tag user 1 donate (C)
  14. 14. Basin areas (numerical) (1, 1) (-1, -1) μ phenotype -1 no donate (D) 0 (0, -1) (-1, 0) (0, 0) q tag user 1 donate (C)
  15. 15. Best response (continuous tag) • Any μi = μ is ESS if bq>c • If μi is uniformly distributed, bq − c µ opt = , ( b − c) q µ opt = 0, ( bq > c ) ( bq ≤ c ) optimal μ b/c=4 2 1.2 q
  16. 16. Numerical simulations noiseless case noisy case μ n = 800 b=1 c = 0.3 q
  17. 17. Conclusions • Tag-based indirect reciprocity is viable when publicity of tolerance is intermediate. – Large publicity → cooperation prevails – Small publicity → defection prevails • Future work: network version?

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