Consider example T > R > P > S
description
Transcript of Consider example T > R > P > S
1
+R+R +S
+T
+T+S
+P+P
Consider example T > R > P > S
Agents try to maximize payoff
Solution := no agent can increase payoff through unilateral change of strategy. E.g., D-vs.-D (T > R and P > S).
Each agent obtains less-than-maximum payoff (P < T) owing to other agent’s adoption of strategy D
Rationality
Nash equilibrium
0
pD
1
t
Consider example T > R > P > S
T, R, P, and S are cell-replication coefficients associated with pairwise collisions
Stable homogeneous steady state, i.e. pD → 1 because T > R and P > S.
Enriching in D reduces fitness of both cell types (because T > P and R > S)
Replicators with fitness
ESS
Evolutionary dynamics providing insight into a related game theory model
Game theory
Prisoner’s dilemma
Tantalizing connections in game theory
Fortune cookie
You
𝑓 0𝛼
𝑅𝛽
𝑆𝛽
+R+R +S
+T
+T+S
+P+P
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Connections: Mechanistic model and quantitative reasoning
$$
$
Other cell
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
3
Population dynamics with table of progeny numbers
+R +S
∆𝐶𝑌𝑂𝑈=𝑓 0𝛼 𝛼∆ 𝑡+ 𝑅𝛽 𝛽∆𝑡 ( 𝐶𝑁 )+𝑆𝛽 𝛽∆ 𝑡 ( 𝐷𝑁 )+𝑂 (∆ 𝑡2 )
𝑆𝛽𝑓 0
𝛼
𝑅𝛽
You
Other cell 𝐶→𝐶+∆𝐶 ;𝐷→𝐷+∆𝐷 ;𝑁→𝑁+∆𝑁𝑡→ 𝑡+∆𝑡 ;
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
∆𝐶𝑌𝑂𝑈=𝑓 0𝛼 𝛼∆ 𝑡+ 𝑅𝛽 𝛽∆𝑡 ( 𝐶𝑁 )+𝑆𝛽 𝛽∆ 𝑡 ( 𝐷𝑁 )+𝑂 (∆ 𝑡2 )
4
+R +S
𝑆𝛽
𝑅𝛽
𝑓 0𝛼
Population dynamics with table of progeny numbersYo
u
Other cell 𝐶→𝐶+∆𝐶 ;𝐷→𝐷+∆𝐷 ;𝑁→𝑁+∆𝑁𝑡→ 𝑡+∆𝑡 ;
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
∆𝐶𝑌𝑂𝑈=𝑓 0𝛼 𝛼∆ 𝑡+ 𝑅𝛽 𝛽∆𝑡 ( 𝐶𝑁 )+𝑆𝛽 𝛽∆ 𝑡 ( 𝐷𝑁 )+𝑂 (∆ 𝑡2 )
5
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
+R +S
𝑆𝛽𝑓 0
𝛼
𝑅𝛽
Population dynamics with table of progeny numbersYo
u
Other cell 𝐶→𝐶+∆𝐶 ;𝐷→𝐷+∆𝐷 ;𝑁→𝑁+∆𝑁𝑡→ 𝑡+∆𝑡 ;
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
Other cell
𝑂 (∆ 𝑡 2 )≔stuff ∆ 𝑡2+stuff ∆ 𝑡 3+stuff ∆ 𝑡4+⋯
∆𝐶𝑌𝑂𝑈=𝑓 0𝛼 𝛼∆ 𝑡+ 𝑅𝛽 𝛽∆𝑡 ( 𝐶𝑁 )+𝑆𝛽 𝛽∆ 𝑡 ( 𝐷𝑁 )+𝑂 (∆ 𝑡2 )
𝐶∆𝐶𝑌𝑂𝑈=[ 𝑓 0+𝑅( 𝐶𝑁 )+𝑆( 𝐷𝑁 )+𝑂 (∆ 𝑡 )]𝐶∆𝑡∆𝐶−𝑂 (∆ 𝑡2 )Yo
u
+R+R +S
+S
𝑓 0𝛼
𝑅𝛽
𝑆𝛽
6
Population dynamics with table of progeny numbers
𝐶→𝐶+∆𝐶 ;𝐷→𝐷+∆𝐷 ;𝑁→𝑁+∆𝑁𝑡→ 𝑡+∆𝑡 ;
(Purple “stuff” need not be same as blue “stuff”)𝑂 (∆ 𝑡 2 )≔stuff ∆ 𝑡2+stuff ∆ 𝑡 3+stuff ∆ 𝑡4+⋯
𝐶→𝐶+∆𝐶 ;𝐷→𝐷+∆𝐷 ;𝑁→𝑁+∆𝑁𝑡→ 𝑡+∆𝑡 ;Other cell
𝑂 (∆ 𝑡 2 )≔stuff ∆ 𝑡2+stuff ∆ 𝑡 3+stuff ∆ 𝑡4+⋯
∆𝐶𝑌𝑂𝑈=𝑓 0𝛼 𝛼∆ 𝑡+ 𝑅𝛽 𝛽∆𝑡 ( 𝐶𝑁 )+𝑆𝛽 𝛽∆ 𝑡 ( 𝐷𝑁 )+𝑂 (∆ 𝑡2 )
𝐶∆𝐶𝑌𝑂𝑈=[ 𝑓 0+𝑅( 𝐶𝑁 )+𝑆( 𝐷𝑁 )+𝑂 (∆ 𝑡 )]𝐶∆𝑡∆𝐶−𝑂 (∆ 𝑡2 )Yo
u
+R+R +S
+S
𝑓 0𝛼
𝑅𝛽
𝑆𝛽
Fitness of C
∆𝐶∆ 𝑡 =[ 𝑓 0+𝑅 (𝐶𝑁 )+𝑆( 𝐷𝑁 )+𝑂 (∆ 𝑡 )]𝐶+𝑂 (∆ 𝑡 )
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Population dynamics with table of progeny numbers𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
Fitness of D
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
∆𝐶−𝑂 (∆ 𝑡2 )𝐶∆𝐶𝑌𝑂𝑈=[ 𝑓 0+𝑅( 𝐶𝑁 )+𝑆( 𝐷𝑁 )+𝑂 (∆ 𝑡 )]𝐶∆𝑡
𝑂 (∆ 𝑡 2 )≔stuff ∆ 𝑡2+stuff ∆ 𝑡 3+stuff ∆ 𝑡4+⋯
∆𝐶𝑌𝑂𝑈=𝑓 0𝛼 𝛼∆ 𝑡+ 𝑅𝛽 𝛽∆𝑡 ( 𝐶𝑁 )+𝑆𝛽 𝛽∆ 𝑡 ( 𝐷𝑁 )+𝑂 (∆ 𝑡2 )
8
Population dynamics with table of progeny numbers
+R+R +S
+T
+T+S
+P+P
𝑓 0𝛼
𝑅𝛽
𝑆𝛽
You
∆𝐶∆ 𝑡 =[ 𝑓 0+𝑅 (𝐶𝑁 )+𝑆( 𝐷𝑁 )+𝑂 (∆ 𝑡 )]𝐶+𝑂 (∆ 𝑡 )
Other cell 𝐶→𝐶+∆𝐶 ;𝐷→𝐷+∆𝐷 ;𝑁→𝑁+∆𝑁𝑡→ 𝑡+∆𝑡 ;
𝑓 0𝛼
𝑇𝛽
𝑃𝛽
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
𝑅𝛽
𝑆𝛽
Other cell
You
9
𝑓 0𝛼
𝑅𝛽
𝑆𝛽
+R+R +S
+T
+T+S
+P+P
Evolution resulting from repeated games
Part
ner 1
Partner 2
+R+R +S
+T
+T+S
+P+P
$ $$
Evolutionary game theory
Game theory
𝑑𝐶𝑑𝑡 =[ 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 ]𝐶
𝑑𝐷𝑑𝑡 =[ 𝑓 0+𝑇 𝑝𝐶+𝑃 𝑝𝐷 ]𝐷
Fitness of C
Fitness of D
10
Quantitative reasoning
Cell population eventually denim rich Both agents choose denim strategy
$$
$
Population dynamics Business payoff analysisWhat propositions might we model? How might conclusions depend on our propositions?
Proposition 1: Consequences depend on social context
Proposition 2: Strategy decisions based on social context
Yes Yes
No YesSloppy guess: Similarities not expected in conclusions for Pr. 1 vs. Pr. 1 and Pr. 2
+R+R +S
+T
+T
+S
+P
+P
+R+R +S
+T
+T
+S
+P
+P
?
Recall prisoner’s dilemma examples (T > R > P > S): Denim is eventually prevalent.
?
$$
$
Cell population eventually denim rich Both agents choose denim strategy
What propositions might we model? How might conclusions depend on our propositions?
Proposition 1: Consequences depend on social context
Proposition 2: Strategy decisions based on social context
Yes Yes
No YesSloppy guess: Similarities not expected in conclusions for Pr. 1 vs. Pr. 1 and Pr. 2
+R+R +S
+T
+T
+S
+P
+P
+R+R +S
+T
+T
+S
+P
+P
?
Recall prisoner’s dilemma examples (T > R > P > S): Denim is eventually prevalent.
Quantitative reasoning
?
11
Population dynamics Business payoff analysis
$$
$
Cell population eventually denim rich Both agents choose denim strategy
What propositions might we model? How might conclusions depend on our propositions?
Proposition 1: Consequences depend on social context
Proposition 2: Strategy decisions based on social context
Yes Yes
No YesSloppy guess: Similarities not expected in conclusions for Pr. 1 vs. Pr. 1 and Pr. 2
+R+R +S
+T
+T
+S
+P
+P
+R+R +S
+T
+T
+S
+P
+P
?
Recall prisoner’s dilemma examples (T > R > P > S): Denim is eventually prevalent.
Quantitative reasoning
?
12
Repetition of Pr. 1 can yield conclusions that seem to have “similarity” with applying Pr. 1 and Pr. 2 once. Beware that time can compensate for lack of thinking.
Population dynamics Business payoff analysis
Cell population eventually denim rich Both agents choose denim strategy
What propositions might we model? How might conclusions depend on our propositions?
Proposition 1: Consequences depend on social context
Proposition 2: Strategy decisions based on social context
Yes Yes
No YesSloppy guess: Similarities not expected in conclusions for Pr. 1 vs. Pr. 1 and Pr. 2
Recall prisoner’s dilemma examples (T > R > P > S): Denim is eventually prevalent.
?
+R+R +S
+T
+T
+S
+P
+P
+R+R +S
+T
+T
+S
+P
+P
Repetition of Pr. 1 can yield conclusions that seem to have “similarity” with applying Pr. 1 and Pr. 2 once. Beware that time can compensate for lack of thinking. 13
Quantitative reasoning
Population dynamics Business payoff analysis
$$
$?
You
𝑑𝐶𝑑𝑡 =( 𝑓 0+𝑅𝑝𝐶+𝑆𝑝𝐷 )𝐶
𝑑𝐷𝑑𝑡 =( 𝑓 0+𝑇 𝑝𝐶+𝑃𝑝𝐷 )𝐷
Fitness of C
Fitness of D 𝑓 0𝛼
𝑅𝛽
𝑆𝛽
+R+R +S
+T
+T+S
+P+P
14
Connections: Mechanistic model and quantitative reasoning
Other cell
$$
$