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

2

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 )

7

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

$$

$