Tantalizing connections in game theory
pD
Evolutionary dynamics providing insight
into a related game theory model
1
+R
+R
+T
+S
+S
0
Consider example T > R > P > S
t
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)
+T
+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
1
Connections: Mechanistic model and quantitative reasoning
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
Other cell
You
+R
+R
+T
+S
+S
+T
$
$
$
+P
+P
2
Population dynamics with table of progeny numbers
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
Other cell
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
𝑡 → 𝑡 + ∆𝑡; 𝐶 → 𝐶 + ∆𝐶; 𝐷 → 𝐷 + ∆𝐷; 𝑁 → 𝑁 + ∆𝑁
You
∆𝐶𝑌𝑂𝑈 =
+R
𝑓0
𝑅
𝐶
𝑆
𝐷
𝛼∆𝑡 + 𝛽∆𝑡
+ 𝛽∆𝑡
+ 𝑂 ∆𝑡 2
𝛼
𝛽
𝑁
𝛽
𝑁
+S
3
Population dynamics with table of progeny numbers
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
Other cell
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
𝑡 → 𝑡 + ∆𝑡; 𝐶 → 𝐶 + ∆𝐶; 𝐷 → 𝐷 + ∆𝐷; 𝑁 → 𝑁 + ∆𝑁
You
∆𝐶𝑌𝑂𝑈 =
+R
𝑓0
𝑅
𝐶
𝑆
𝐷
𝛼∆𝑡 + 𝛽∆𝑡
+ 𝛽∆𝑡
+ 𝑂 ∆𝑡 2
𝛼
𝛽
𝑁
𝛽
𝑁
+S
4
Population dynamics with table of progeny numbers
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
Other cell
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
𝑡 → 𝑡 + ∆𝑡; 𝐶 → 𝐶 + ∆𝐶; 𝐷 → 𝐷 + ∆𝐷; 𝑁 → 𝑁 + ∆𝑁
You
∆𝐶𝑌𝑂𝑈 =
+R
𝑓0
𝑅
𝐶
𝑆
𝐷
𝛼∆𝑡 + 𝛽∆𝑡
+ 𝛽∆𝑡
+ 𝑂 ∆𝑡 2
𝛼
𝛽
𝑁
𝛽
𝑁
+S
5
Population dynamics with table of progeny numbers
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
Other cell
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
𝑡 → 𝑡 + ∆𝑡; 𝐶 → 𝐶 + ∆𝐶; 𝐷 → 𝐷 + ∆𝐷; 𝑁 → 𝑁 + ∆𝑁
𝑓0
𝑅
𝐶
𝑆
𝐷
∆𝐶𝑌𝑂𝑈 = 𝛼∆𝑡 + 𝛽∆𝑡
+ 𝛽∆𝑡
+ 𝑂 ∆𝑡 2
𝛼
𝛽
𝑁
𝛽
𝑁
𝑂 ∆𝑡 2 ≔ stuff ∆𝑡 2 + stuff∆𝑡 3 + stuff∆𝑡 4 + ⋯
You
+R
+R
+S
+S
𝐶∆𝐶𝑌𝑂𝑈 = 𝑓0 + 𝑅
𝐶
𝐷
+𝑆
+ 𝑂 ∆𝑡 𝐶∆𝑡
𝑁
𝑁
∆𝐶 − 𝑂 ∆𝑡 2
𝑂 ∆𝑡 2 ≔ stuff ∆𝑡 2 + stuff∆𝑡 3 + stuff∆𝑡 4 + ⋯
(Purple “stuff” need not be same as blue “stuff”)
6
Population dynamics with table of progeny numbers
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
Other cell
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
𝑡 → 𝑡 + ∆𝑡; 𝐶 → 𝐶 + ∆𝐶; 𝐷 → 𝐷 + ∆𝐷; 𝑁 → 𝑁 + ∆𝑁
𝑓0
𝑅
𝐶
𝑆
𝐷
∆𝐶𝑌𝑂𝑈 = 𝛼∆𝑡 + 𝛽∆𝑡
+ 𝛽∆𝑡
+ 𝑂 ∆𝑡 2
𝛼
𝛽
𝑁
𝛽
𝑁
𝑂 ∆𝑡 2 ≔ stuff ∆𝑡 2 + stuff∆𝑡 3 + stuff∆𝑡 4 + ⋯
You
+R
+R
+S
+S
𝐶∆𝐶𝑌𝑂𝑈 = 𝑓0 + 𝑅
𝐶
𝐷
+𝑆
+ 𝑂 ∆𝑡 𝐶∆𝑡
𝑁
𝑁
∆𝐶 − 𝑂 ∆𝑡 2
∆𝐶
𝐶
𝐷
= 𝑓0 + 𝑅
+𝑆
+ 𝑂 ∆𝑡 𝐶 + 𝑂 ∆𝑡
∆𝑡
𝑁
𝑁
7
Population dynamics with table of progeny numbers
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
Other cell
You
+R
+R
+T
+S
+S
+T
+P
+P
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
𝑡 → 𝑡 + ∆𝑡; 𝐶 → 𝐶 + ∆𝐶; 𝐷 → 𝐷 + ∆𝐷; 𝑁 → 𝑁 + ∆𝑁
𝑇𝐶
𝑓0
𝑅
𝑆
𝐷
∆𝐶𝑌𝑂𝑈 = 𝛼∆𝑡 + 𝑅𝛽∆𝑡
+ 𝛽∆𝑡
+ 𝑂 ∆𝑡 2
𝛽𝑁
𝛼
𝛽
𝛽
𝑁
𝑂 ∆𝑡 2 ≔ stuff𝛽∆𝑡 2 + stuff∆𝑡 3 + stuff∆𝑡 4 + ⋯
𝑃
𝐶 𝑓0
𝐷
𝑆
𝐶∆𝐶𝑌𝑂𝑈 = 𝑓0 + 𝑅
+𝑆
+ 𝑂 ∆𝑡𝛽 𝐶∆𝑡
𝑁𝛼
𝑁
𝛽
2
∆𝐶 − 𝑂 ∆𝑡
∆𝐶
𝐶
𝐷
= 𝑓0 + 𝑅
+𝑆
+ 𝑂 ∆𝑡 𝐶 + 𝑂 ∆𝑡
∆𝑡
𝑁
𝑁
8
Evolution resulting from repeated games
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
Partner 2
Other cell
$
$
$
+R
+S
+S
+T
+R
+T
+P
+P
Partner 1
You
+R
+R
+T
+S
+S
+T
+P
+P
9
Quantitative reasoning
What propositions might we model? How might conclusions depend on our propositions?
Population dynamics
Business payoff analysis
Yes
Proposition 1: Consequences depend on social context
Yes
$
+R
+R
+T
+S
+S
+T
No
$
+P
+P
?
+R
+R
$
+T
+S
+S
+T
+P
+P
Proposition 2: Strategy decisions based on social context
Yes
Sloppy 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.
Cell population eventually denim rich
Both agents choose denim strategy
?
10
Quantitative reasoning
What propositions might we model? How might conclusions depend on our propositions?
Population dynamics
Business payoff analysis
Yes
Proposition 1: Consequences depend on social context
Yes
$
+R
+R
+T
+S
+S
+T
No
$
+P
+P
?
+R
+R
$
+T
+S
+S
+T
+P
+P
Proposition 2: Strategy decisions based on social context
Yes
Sloppy 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.
Cell population eventually denim rich
Both agents choose denim strategy
?
11
Quantitative reasoning
What propositions might we model? How might conclusions depend on our propositions?
Population dynamics
Business payoff analysis
Yes
Proposition 1: Consequences depend on social context
Yes
$
+R
+R
+T
+S
+S
+T
No
$
+P
+P
?
+R
+R
$
+T
+S
+S
+T
+P
+P
Proposition 2: Strategy decisions based on social context
Yes
Sloppy 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.
Cell population eventually denim rich
Both agents choose denim strategy
?
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.
12
Quantitative reasoning
What propositions might we model? How might conclusions depend on our propositions?
Population dynamics
Business payoff analysis
Yes
Proposition 1: Consequences depend on social context
Yes
$
+R
+R
+T
+S
+S
+T
No
$
+P
+P
?
+R
+R
$
+T
+S
+S
+T
+P
+P
Proposition 2: Strategy decisions based on social context
Yes
Sloppy 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.
Cell population eventually denim rich
Both agents choose denim strategy
?
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
Connections: Mechanistic model and quantitative reasoning
𝑑𝐶
= 𝑓0 + 𝑅𝑝𝐶 + 𝑆𝑝𝐷 𝐶
𝑑𝑡
Fitness of C
𝑑𝐷
= 𝑓0 + 𝑇𝑝𝐶 + 𝑃𝑝𝐷 𝐷
𝑑𝑡
Fitness of D
𝑅
𝛽
𝑆
𝛽
𝑓0
𝛼
Other cell
You
+R
+R
+T
+S
+S
+T
$
$
$
+P
+P
14