The good side of running away

The good side of running away Introducing signalling into Conways Game of Life Simon Schulz si.schulz@student.uni-tuebingen.de 20. Januar 2013

Overview Introduction How to improve the game The GOLS Game Implementation Results

Introduction

Conways Game of Life Is a cellular automaton, which follows four rules:

Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population).

Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation.

Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation. 3 Any live cell with more than three live neighbours dies(overcrowding).

Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation. 3 Any live cell with more than three live neighbours dies(overcrowding). 4 Any dead cell with exactly three live neighbours becomes a live cell (reproduction)

Conways Game of Life Is a cellular automaton, which follows four rules: 1 Any live cell with fewer then two living neighbours dies (under-population). 2 Any live cell with two or three live neighbours lives on to the next generation. 3 Any live cell with more than three live neighbours dies(overcrowding). 4 Any dead cell with exactly three live neighbours becomes a live cell (reproduction)

Conways Game of Life Rules applied simultaneously Zero player game, seed at beginning Deterministic

Conways Game of Life Rules applied simultaneously Zero player game, seed at beginning Deterministic Gardner, Martin (1970/10) Because of Life s analogies with the rise, fall and alterations of a society of living organisms, it belongs to a growing class of what are called simulation games

Point of view Commonly on cellular level

Point of view Commonly on cellular level But macro view:

Point of view Commonly on cellular level But macro view: Cells as cities in a given area

Point of view Commonly on cellular level But macro view: Cells as cities in a given area Relation of available resources...

Point of view Commonly on cellular level But macro view: Cells as cities in a given area Relation of available resources...... and e.g. trading.

Point of view Commonly on cellular level But macro view: Cells as cities in a given area Relation of available resources...... and e.g. trading. Communication

Communication Requires some intelligence Intelligence gives ability to make decisions

Communication Requires some intelligence Intelligence gives ability to make decisions Is there any decision that increases amount of surviving cities?

How to improve the game

Case studies Allow action between generations What are reasonable actions?

Case studies Allow action between generations What are reasonable actions? Influence on the rules Influence on the global survivability

Case studies Allow action between generations What are reasonable actions? Influence on the rules Influence on the global survivability What s about ABANDONMENT?

4-die-modification vs. normal game 2 4die/basic game 1.5 ratio 4die/basic 1 0.5 0 50 runs

4-die-modification vs. normal game 1600 Long term behaviour 4die 1400 1200 Number living cells 1000 800 600 400 200 0 0 200 400 600 800 1000 Number of Generations

Use that knowledge to...... give the cells the ability to abandon Depending on signalling Preplay communication Step before basic GoL rule appliance

Use that knowledge to...... give the cells the ability to abandon Depending on signalling Preplay communication Step before basic GoL rule appliance Example: citizens can decide to abandon the area e.g. when resources run short

The signalling system

Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }.

Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }. Definition: Sender & Receiver The sender is a cell c s. The receiver is a cell c r, c s c r, S, R C = {c i : c i Living cells}

Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }. Definition: Sender & Receiver The sender is a cell c s. The receiver is a cell c r, c s c r, S, R C = {c i : c i Living cells} Definition: Action An action is the decision to die (a 2 ) or not to die (a 1 ), before the next game-based selection process begins, A = {a 1, a 2 }.

Definitions Definition: State For a given cell c i the state is t n, where n = 0,..., 8, number of living cells around c i. The set of states is therefore defined as T = {t 0,..., t 8 }. Definition: Sender & Receiver The sender is a cell c s. The receiver is a cell c r, c s c r, S, R C = {c i : c i Living cells} Definition: Action An action is the decision to die (a 2 ) or not to die (a 1 ), before the next game-based selection process begins, A = {a 1, a 2 }. Definition: Messages A message is a signal that can be choosen from := {M 0,..., M 8 }.

Utility-function The utility is calculated by u(c r, a i ) = N(c r, t + 1, a i ) N(c r, t) where c r is a receiver cell a i is a specific action t is the time N is the neighbourhood function

The signalling game Definition: Signalling game for Conways GoL G = (S, R, T, M, A, u)

GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ)

GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ) Where G is the game

GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ) Where G is the game ϕ selects the sender & receiver

GOLS Game Definition: GOLS Game SS = (G, ϕ, Pr m, Pr a, µ) Where G is the game ϕ selects the sender & receiver µ is the update dynamic

Selection functions

Selection functions Definition of Pr m Pr m : T M

Selection functions Definition of Pr m Pr m : T M Definition of Pr a Pr a : T M M

Agent selection function Definition: Agent selection function C = (Cell 1,..., Cell n ) ϕ(c) = {(Cell i, Cell j )} ϕ(c \ {Cell i, Cell j }) where i, j N, i, j < n, i random, j such that Cell j random neighbour of Cell i

Update dynamics Definition: Update function Bush-Mosteller-Reinforcement µ(x) = pr old (x) + α (1 pr old (x)) α = lp u, where lp is the learning parameter u is the utility gotten

Paste it together (C s, C r ) ϕ(c)

Paste it together (C s, C r ) ϕ(c) T s R m M Tr R a A

Paste it together (C s, C r ) ϕ(c) T s R m M Tr R a A ut = u(c r, A)

Paste it together (C s, C r ) ϕ(c) T s R m M Tr R a A ut = u(c r, A) µ(t s ), µ(m, T r )

Conclusion Global knowledge of all cells

Conclusion Global knowledge of all cells Cells only know the neighbours

Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour

Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour... their selves

Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour... their selves Action based on local knowledge

Conclusion Global knowledge of all cells Cells only know the neighbours... of a random neighbour... their selves Action based on local knowledge Communication on a random base

Implementation

Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast)

Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast) Time depended strategies: High density vs few left

Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast) Time depended strategies: High density vs few left Chaos - Difficult to measure

Problems that occured Testing showed... Poor signalling gets punished hard and fast (Populations die out fast) Time depended strategies: High density vs few left Chaos - Difficult to measure Utility choice of utility function

Solutions Solution for hard punishment Repeat update progress n-times Large initial populations

Solutions Solution for hard punishment Repeat update progress n-times Large initial populations Solution for time-depended strategies Bush-Mosteller reinforcement for faster learning Roth-Erev reinforcement too slow Repeat update progress n-times

Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average

Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1

Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1 And then build the ratio r = c signalling c basic

Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1 And then build the ratio r = c signalling c basic This ratio is used as a measurement for the survivability.

Solutions Solution for measuring the chaos Adequate amount of runs Sum up the cell count and calculate average c = 1 t n CellsNum(t) t=1 And then build the ratio r = c signalling c basic This ratio is used as a measurement for the survivability.

Solutions Solution for finding a fairly good utility function Number of global cells not representative Again... chaos Commonly number of cells shrinking Examining all constellations too complex Solution already introduced function

Demonstration

Results

Quantitative evaluation Grid size = 70 70 Random cells p C =.25 modification vs basic 5 repetitions of signalling before each round actions in 5-th round

With a given meaning 2 signalling nm/basic game ratio signalling nm/basic 1.5 1 0.5 0 100 runs

With a no given meaning 2 signalling wm/basic game ratio signalling wm/basic 1.5 1 0.5 0 100 runs

Results compared Runs Successful Average 4-die 50 100% 136% given meaning 100 83% 119% no giv. meaning 100 77% 114%

Long term with a given meaning 2500 Long term behaviour signaling with given meaning 2000 Number living cells 1500 1000 500 0 0 200 400 600 800 1000 Number of generations

Long term without a given meaning 2500 Long term behaviour signaling with no given meaning 2000 Number living cells 1500 1000 500 0 0 200 400 600 800 1000 Number of generations

Qualitative evaluation Signalling strong when huge population Big grid size Important to get a good strategy fast Weak when only a few left Wrong strategy hard punishment

Final statement Taking Conways Game of Life as a metaphor for life, one could say Signalling can achieve higher survivability Signalling can successfully evolve in a high-punishing environment Poor signalling is deadly in a high-punishing environment