Games on Cellular Spaces: An Evolutionary Approach

P. R. Andrade, A. M. V. Monteiro, G. Camara

During the first four billion years or life on Earth, the primary means of information transfer was genetic. (M. Nowak)

EPIA/SSM 2009

Reviewer 1

Summary of paper	This paper studies the conditions where a population of agents reaches equilibrium in evolutionary game theory. An agent-based model is presented, where individuals compete for space using mixed strategies. The simulation of this model shows that the population’s mean strategy always converges to a stable state, close and above to the analytic equilibrium.
Relevance to the track	Relevant to the Track
Comments on relevance to the track	This paper is relevant to the ABM issue “Agent-based experimental economics”.
Originality	Same thing has been done before
Comments on originality	It is not clear the originality of the approach. The state of the art concerns mainly the description of theoretical background in non-cooperative games, Nash equilibrium and Evolutionary Stable Strategies. However, it would have been interesting to reference other work, proposing some Social Simulation approach complementary to the analytic studies in Evolutionary Game Theory.
Literature awareness	Good but missing a few references
Comments on literature awareness	See previous comment.
Scientific/technical soundness	Basically okay
Significance of results	Paper will moderately influence researchers close to the area
Comments on significance of results	Clearly interesting approach for Evolutionary Game Theory researchers.
Clarity of presentation	Basically okay
Overall rating	Accept
Reviewer confidence	I'm moderately familiar with this area
Detailed comments	Section 1, page 2: “Memorizing the last results is not practicable within this model, since the agent may not have the same opponent in its next confront.” - however, agents could have been designed to memorize and apply the well succeeded strategies, whoever the opponents are, improving this way their satisfaction. Section 5, 1st paragraph: “The parameters chosen were: 3 as the number of descendants, 0.1% as the chance of mutation, and ±0.1 as the change in the inherited strategy, with 50% of probability of each, once the mutation is activated.” - why these initial parameters? It lacks some explanation about these choices.

Reviewer 2

Summary of paper	The paper studies a model of evolutionary games on a grid. Agents compete when they are located in the same cell, through a chicken game. Each agent has a strategy defined by its probability to escalate or not. When the satisfaction of an agent is below a threshold, it moves to a random neighbouring cell. When the satisfaction of the agent is below another threshold, the agent leaves the game. The agents which are still in the game after a given number of time steps create several offspring which inherit the strategy of their father, with some mutation rate. The simulations show that the stationary state of the system include several strategies, and that the diversity of these strategies is higher when the mutation rate is high. This result is much richer than the theoretical stationary state in the standard game which includes only one strategy.
Relevance to the track	Relevant to the Track
Comments on relevance to the track	I think that the paper fits well the Social Simulation and Modelling track
Originality	A direct extension of existing work
Literature awareness	Clear specification of relation to rest of field
Scientific/technical soundness	Basically okay
Comments on scientific/technical soundness	I think the paper is globally technically sound. The simulations are convincing. See global comments for more details.
Significance of results	Paper will moderately influence researchers close to the area
Clarity of presentation	Basically okay
Comments on clarity of presentation	Maybe some pesudocode of the basic dynamics would complete well the description. There are a few typos and bugs remaining in the text.
Overall rating	Accept
Reviewer confidence	I'm very familiar with this area
Detailed comments	Interesting paper with clear results. Two criticisms however: - I think it is probably possible to derive an analytical model which predicts the proportion of strategies at the stationary state. To do this, you need to write the master equation ruling the fluxes between the different strategies, and I don't see any major difficulty for getting at least a good approximation of this. Thus the affirmation that the result is only achievable through explicit simulation seems a bit dangerous to me.- the description of the local and global satisfactions, and their evolution lacks precision. Minor remark: In section 2, I don't understand the difference between pa (pb) and sa (sb).

Reviewer 3

Summary of paper	The paper presents an evolutionary extension of a previous work on a game played in a grid. The paper gives preliminary results on the experimentation showing a convergence towards the theoretical equilibrium in the context of the game when playing the 'chicken' payoff matrix.
Relevance to the track	Relevant to the Track
Comments on relevance to the track	The paper presents a number of experiments in full detail. It is a simulation of behaviour although the
Originality	A direct extension of existing work
Comments on originality	The results are rather straightforward and largely result of the parameters choses. e.g. +-0.1 in the mutation determines how close the curve is to the theoretical equilibrium. If using the complete [0,1] range, the simulations would have shown an almost perfect equilibrium.
Literature awareness	Clear specification of relation to rest of field
Scientific/technical soundness	Basically okay
Significance of results	Paper will moderately influence researchers close to the area
Clarity of presentation	Beautifully clear
Comments on clarity of presentation	Very nicely written paper.
Overall rating	Neutral
Reviewer confidence	I'm moderately familiar with this area
Detailed comments	The paper is a natural extension of the basic model allowing for an evolutionary approach very similar to the extansion that Axelrod did over the basic game to make it evolutionary. The paper is in that respect correct. However the results are very straightforward and differently from Axelrod's results, the experiments basically show that the game tends to the equilibrium. The fitness function 'determines' the solution and in that respect there is not much to be learned from the paper. The suggested changes for future work as expressed in the conclusion would not produce any non-expected result. The main conclusion that the convergence is 'over' the equilibrium is an artefact of the parameters used.

SNAMAS 2009

Review 1

OVERALL RATING: -3 (strong reject)
REVIEWER'S CONFIDENCE: 3 (high)
Originality: 2 (poor)
Relevance: 2 (poor)
Technical soundness: 3 (fair)
Significance: 2 (poor)
Quality of presentation: 3 (fair)

The central issue of this paper seems to be the addition of “space” as a concept to a model of non-cooperative games. The issue of “space” is discussed in many simulation studies, even those using game theory or cellular automata so th enovelty of the approach is very limited. The use of “the space” instead of space is a linguistic error that irritates as are several other problems with language. My main point however is the absence of a relation between this work and the issue of the workshop namely social networks. The paper mentions no relationaship between physical positions and social networks and doe snot consider other ties than those with strength 0 (not a neighbour) or 1 (a neighbour), no direction in relationships is acknownledged nor is there a sense of a network (ie consisting of vaiours nodes with relationships, strenghts and directions) that needs analysis.

Review 2

OVERALL RATING: -1 (weak reject)
REVIEWER'S CONFIDENCE: 4 (expert)
Originality: 3 (fair)
Relevance: 1 (very poor)
Technical soundness: 3 (fair)
Significance: 3 (fair)
Quality of presentation: 4 (good)

The paper deals with evolutionary games and agent based simulation. While mathematically demonstrable under ideal hypotheses, the paper shows, through repeated simulations, that a Nash Equilibrium is reachable even when the assumptions are relaxed and, in particular, agent based simulation can show the step by step dynamics of the system, while mathematical model can’t. Pros: In general the paper is well written and its goals are immediately understandable. It fits the conference topic. Cons: While I agree that agent based simulation can show the dynamics of a system, while mathematical models can’t, I think that the conclusions driven by the authors where quite straightforward. Rather than reaching an equilibrium, I’d refer to the results as a “survival of the fittest”. In fact, through evolution, the agents are substituted by some descendants with a behavior which is like the parental one, plus or minus a mutation. Since it’s obvious that an agent with a better strategy wins over one with e worse one, by repeating the game many times with random agents the ones with worse strategies will of course diminish their energy and eventually disappear. Besides the simulation features many random distributions and probabilities, which are not described in detail, that can highly impact the final results. Were the experiments repeated many times and normalized? This is not clear from the paper. Finally, the graphs are not easily readable and the tag “figure 2” is given to two figures in the paper.

This paper can be accepted only with big modifications. While the proposed goal is interesting, I think that the model is too simple and above all features too many stochastic points. Besides, I think it would be interesting to understand why agents with a sub-optimal strategy survive (s(X+1)/10 and s(X+2)/10) and not just showing that this happens.

S. E. Riechert

Games spiders play: behavioural variability in territorial disputes. Behavioural Ecology and Sociobiology 3:135-162. 1979.
Games spiders play II: Resource assessment strategies. Behavioural Ecology and Sociobiology 6:121-128. 1979.
The consequences of being territorial: spiders, a case of study. American Naturalist 117:871-892. 1981.
Spider interaction and strategies: communication vs. coercion. 281-315. In Spider communication, Mechanisms and ecological significance. Princeton University Press. 1982.

Adaptive dynamics in space

Lion, Sebastien

Over the last twenty years, the role of spatial self-structuring as a template for evolution has drawn much attention. Spatial structure can be an important component of the eco-evolutionary feedback loop: the evolution of a trait (e.g. migration) can shape the local structure of the population, which in turn creates new selective pressures on the evolving trait. As a consequence, the evolution of spatially structured populations often displays very different features from the evolution of well-mixed populations.

ESS for Chicken and Nash Equilibrium

I is stable iff:

E(I,I) > E(J,I)

or

E(I,I) = E(J,I) and E(I,J) > E(J,J)

	S0	S1
S0	0	-1
S1	1	-10

S0 is not ESS against S1:

E(I,I) > E(J,I)
E(S0,S0) > E(S1,S0)
0 > 1 (FALSE)

E(I,I) = E(J,I) and E(I,J) > E(J,J)
E(S0,S0) = E(S1,S0) and E(S0,S1) > E(S1,S1)
0 = 1 (FALSE)

S1 is not ESS against S0:

E(I,I) > E(J,I)
E(S1,S1) > E(S0,S1)
-10 > -1 (FALSE)

E(I,I) = E(J,I) and E(I,J) > E(J,J)
E(S1,S1) = E(S0,S1) and E(S1,S0) > E(S0,S0)
-10 = -1 (FALSE)

Adding Neq (S0.1) to the game, we have:

	S0	S0.1	S1
S0	0	-0.1	-1
S0.1	0.1	-0.1	-1.9
S1	1	-0.1	-10

S0.1 is ESS against S0:

E(I,I) > E(J,I)
E(S0.1,S0.1) > E(S0,S0.1)
-0.1 > -0.1 (FALSE)

E(I,I) = E(J,I) and E(I,J) > E(J,J)
E(S0.1,S0.1) = E(S0,S0.1) and E(S0.1,S0) > E(S0,S0)
-0.1 = -0.1 and +0.1 > 0 (TRUE)

S0.1 is ESS against S1:

E(I,I) > E(J,I)
E(S0.1,S0.1) > E(S1,S0.1)
-0.1 > -0.1 (FALSE)

E(I,I) = E(J,I) and E(I,J) > E(J,J)
E(S0.1,S0.1) = E(S1,S0.1) and E(S0.1,S1) > E(S1,S1)
-0.1 = -0.1 and -1.9 > -10 (TRUE)