greca talk

The story so far...Steven Hamblin

The Hero of our story...

Act I

Producer

Producers

Scrounger

Other forms of scrounging.

50% producer. 50% scrounger.100% 0%

0% 100%producer.

producer.

scrounger.

scrounger.

Rules

Rules• Relative payoff sum

Rules• Relative payoff sum

• Perfect Memory

Rules• Relative payoff sum

• Perfect Memory

• Linear Operator

Relative Payoff Sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative Payoff Sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative Payoff Sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative Payoff Sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Relative Payoff Sum

where 0 < x < 1 is a memory factor,

ri > 0 is the residual value associated with alternative i,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Perfect Memory

Si(t) = � + Ri(t)/(⇥ + Ni(t))

where Ri(t) is the cumulative payo�s from alternative i to time t,

Ni(t) is the number of time periods from the beginning in which the option

was selected,

� and ⇥ are parameters.

Perfect Memory

Si(t) = � + Ri(t)/(⇥ + Ni(t))

where Ri(t) is the cumulative payo�s from alternative i to time t,

Ni(t) is the number of time periods from the beginning in which the option

was selected,

� and ⇥ are parameters.

Perfect Memory

Si(t) = � + Ri(t)/(⇥ + Ni(t))

where Ri(t) is the cumulative payo�s from alternative i to time t,

Ni(t) is the number of time periods from the beginning in which the option

was selected,

� and ⇥ are parameters.

Perfect Memory

Si(t) = � + Ri(t)/(⇥ + Ni(t))

where Ri(t) is the cumulative payo�s from alternative i to time t,

Ni(t) is the number of time periods from the beginning in which the option

was selected,

� and ⇥ are parameters.

Linear Operator

Si(t) = xSi(t� 1) + (1� x)Pi(t)

where 0 < x < 1 is a memory factor,

Pi(t) is the payo� to alternative i at time t, and

Si(t) is the value that the animal places on the behavioural alternative i at

time t.

Relative Payoff Sum?

Perfect Memory?

Linear Operator?

Bird Start

At a patch with food?

Feed

Produce or scrounge?

Produce Scrounge

Move randomly

Yes

Any conspecifics

feeding?No

Move to closest

Closest still feeding?

There yet?

No

Yes

No

NO

Yes

Bird Start

At a patch with food?

Feed

Produce or scrounge?

Produce Scrounge

Move randomly

Yes

Any conspecifics

feeding?No

Move to closest

Closest still feeding?

There yet?

No

Yes

No

NO

Yes

• 5 or 10 birds.

• Foraging grid is a regular 10x10 grid, with movement in the 4 cardinal directions.

• 20 patches on the grid, with 10 or 20 food items in each.

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Si(t) = � + Ri(t)/(⇥ + Ni(t))

Si(t) = xSi(t� 1) + (1� x)Pi(t)

Relative Payoff Sum?

Perfect Memory?

Linear Operator?

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Si(t) = � + Ri(t)/(⇥ + Ni(t))

Si(t) = xSi(t� 1) + (1� x)Pi(t)

Relative Payoff Sum?

Perfect Memory?

Linear Operator?

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)

Si(t) = � + Ri(t)/(⇥ + Ni(t))

Si(t) = xSi(t� 1) + (1� x)Pi(t)

Relative Payoff Sum?

Perfect Memory?

Linear Operator?

Multiple stable rules with multiple parameters?

Genetic Algorithms

• Algorithms that simulate evolution to solve optimization problems.

Initial population

Measure fitness

Select for

reproduction

Mutation

Exit> n generations

Foraging / Learning rule simulation.

Genetic algorithm to optimize parameters and simulate population dynamics.

Results to date

rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules

02

46

810

Relative Payoff Sum Perfect Memory Linear Operator

0 100

rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules

0200

400

600

800

Relative Payoff Sum Perfect Memory Linear Operator

0 100

rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules rules

050

100

150

200

250

300

350

Relative Payoff Sum Perfect Memory Linear Operator

0 100

Si(t) = xSi(t� 1) + (1� x)ri + Pi(t)Relative Payoff Sum

rp >> rs for large population sizes.

-1 0 1 2 3 4 5 6 7 8

1

2

3

4

5

Producer residual

Scrounger residual

Time without payo! to behaviour

Value assignedto behaviour

What does that mean?

• Under the assumptions of this model, the Relative Payoff Sum rule is optimal.

• Whether RPS is favored depends on payoff variance:

• low variance = more attractive power.

• Differences in residuals gives a prediction for empirical tests.

Next steps?

Other games...

Foraging / Learning rule simulation.

Genetic algorithm to optimize parameters and simulate population dynamics.

Foraging / Learning rule simulation.

Genetic algorithm to optimize parameters and simulate population dynamics.

Genetic programming to optimize rule structure.

Act II

+

Foraging / Learning rule simulation.

Foraging / Learning rule simulation.

Swappable grids (Moore / VN / Hex / Dirichlet)

Foraging / Learning rule simulation.

Genetic algorithm to optimize parameters and simulate population dynamics.

Swappable grids (Moore / VN / Hex / Dirichlet)

Results to date

Act III

+

Node Relationship

Node Relationship

Node Relationship

One field, a few names...

• Graph theory....

• Social network analysis....

• Network theory...

Graph measures...

Graph measures...

• Degree

Graph measures...

• Degree

• Centrality

Graph measures...

• Degree

• Centrality

• Clustering

Graph measures...

• Degree

• Centrality

• Clustering

• Path length

Graph measures...

• Degree

• Centrality

• Clustering

• Path length

• Etc...

Six degrees...

• Small world network:

• High clustering, low path length.

# of connections to other foragers

Birds

# of connections to other foragers

Birds

Most birds have few connections

# of connections to other foragers

Birds

Most birds have few connections

A few birds have many connections

=

=

Foraging / Learning rule simulation.

Small world network analysis

The end of the story.