Marie-Edith Bissey and Guido Ortona (2002)
The Integration of Defectors in a Cooperative Setting
Journal of Artificial Societies and Social Simulation
vol. 5, no. 2
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Received: 10-Dec-2001 Accepted: 17-Mar-2002 Published: 31-Mar-2002
Given an indefinitely repeated non cooperative game, a strategy I is a convention if all the players adopt I and for every player
(a) E(I,I) = E(J,I)
(b) if E(I,I) = E(J,I), E(I,J) > E(J,J).
for every strategy J≠I, where E(a,b) is the expected payoff of a player adopting strategy a against a player adopting strategy b.
(c) There is more than one strategy satisfying conditions (a) and (b).
(a) The simulation is based on a spatial prisoner's dilemma where players meet with their neighbours, play the PD game and move to another part of the space at the end of each round[7]. Players are located in the space according to three specifications:
Table 1: The Space and how Players move in it | ||
The space for the Invasion setting at the beginning: Immigrants are in red in the upper left hand side corner. Natives are in green. All players are disposed randomly on the space. This disposition of players in the space is also the one used in the Ghetto setting. | The space for the Invasion setting after a few rounds of the simulation. Players have moved, and we can see that Immigrants have started to "invade" the Natives' space (and vice versa). In the picture, Immigrants who have learned to cooperate appear in blue. Isolated players who could not be part of any group and are sleeping for the current round are represented in white (Natives) or yellow (Immigrants). | The space for the Invasion setting at the end of the simulation. Both Natives and Immigrants can move anywhere in the space (as in the Random setting). In this case, all immigrant players have learned to cooperate (no more players are red). |
Table 2: The three payoff settings | ||||||
Number of Defectors | First setting payoffs | Second setting payoffs | Third setting payoffs | |||
Defectors | Cooperators | Defectors | Cooperators | Defectors | Cooperators | |
0 | -- | n | -- | n | -- | 2 |
1 | n+1 | n-2 | n+1 | 0 | 3 | 2((n-2)/n-1)) |
i | n-(i-1) | n-(i+1) | n-(i-1) | 0 | 3 | 2((n-(i+1))/(n-1) |
n | 1 | -- | 1 | -- | 1 | -- |
Table 3: Example of punishment for Payoff Setting 2, in a group of size 5 | |||
Number of defectors (size of group: 5) | Payoff defectors (no punishment) | Payoff cooperators | Payoff defectors (with punishment) |
0 | -- | 5 | -- |
1 | 6 | 0 | -1 |
2 | 4 | 0 | -1 |
3 | 3 | 0 | -1 |
4 | 2 | 0 | -1 |
5 | 1 | 0 | 1 |
Table 4: Number of rounds for Immigrants to learn to cooperate (payoff setting 2) | |||||||||||
PAYOFF SETTING 2 | Group Size | ||||||||||
Memory | Population | Position | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
NORMAL | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 2 | 3 | 2 | 3 | 3 | 3 |
1 | 5000 | invasion3 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 |
10 | 5000 | random | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 3 | 3 |
10 | 5000 | invasion3 | 11 | 10 | 10 | 11 | 11 | 11 | 11 | 11 | 11 |
INTELLI | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 5 |
1 | 5000 | invasion3 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 12 | 13 |
10 | 5000 | random | 6 | 5 | 6 | 5 | 5 | 6 | 7 | 16 | 22 |
10 | 5000 | invasion3 | 10 | 12 | 15 | 15 | 18 | 19 | 22 | 24 | 25 |
CLEVER | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
1 | 5000 | invasion3 | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
10 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
10 | 5000 | invasion3 | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
NOBEL | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 |
1 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
10 | 5000 | random | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 |
10 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
Table 5: Number of rounds for Immigrants to learn to cooperate (payoff setting 1) | |||||||||||
PAYOFF SETTING 1 | Group Size | ||||||||||
Memory | Population | Position | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
NORMAL | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
1 | 5000 | invasion3 | 6 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 4 |
10 | 5000 | random | 6 | 6 | 6 | 7 | 6 | 6 | 7 | 7 | 7 |
10 | 5000 | invasion3 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 | 11 |
INTELLI | |||||||||||
1 | 5000 | random | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
1 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
10 | 5000 | random | 6 | 7 | 8 | 8 | 9 | 9 | 11 | 13 | 16 |
10 | 5000 | invasion3 | 10 | 14 | 15 | 18 | 19 | 21 | 22 | 23 | 24 |
CLEVER | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
1 | 5000 | invasion3 | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
10 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
10 | 5000 | invasion3 | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
NOBEL | |||||||||||
1 | 5000 | random | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
1 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
10 | 5000 | random | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
10 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
Table 6: Number of rounds for Immigrants to learn to cooperate (payoff setting 3) | |||||||||||
PAYOFF SETTING 3 | Group Size | ||||||||||
Memory | Population | Position | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
NORMAL | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
1 | 5000 | invasion3 | 6 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 |
10 | 5000 | random | 6 | 9 | 10 | 10 | 12 | 12 | 12 | 13 | 13 |
10 | 5000 | invasion3 | 11 | 12 | 14 | 14 | 15 | 16 | 16 | 15 | 16 |
INTELLI | |||||||||||
1 | 5000 | random | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
1 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
10 | 5000 | random | 6 | 10 | 12 | 13 | 16 | 17 | 19 | 25 | 34 |
10 | 5000 | invasion3 | 10 | 16 | 21 | 23 | 26 | 29 | 30 | 35 | 35 |
CLEVER | |||||||||||
1 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
1 | 5000 | invasion3 | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
10 | 5000 | random | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
10 | 5000 | invasion3 | 6 | 5 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |
NOBEL | |||||||||||
1 | 5000 | random | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
1 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
10 | 5000 | random | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
10 | 5000 | invasion3 | 6 | 6 | 7 | 7 | 8 | 8 | 9 | 10 | 10 |
sizeOfPopulation: total number of players in the game (an integer), fixed at 5000
sizePDGroups: size of the Prisonner's Dilemma groups (an integer, smaller than sizeOfPopulation), varied from 2 (the smallest possible) to 10
proportionImmigrants: total proportion of immigrants in the game (a number between 0 and 1), fixed at 5%
playerMover: how players are positionned in the space at the beginning of the game and how they move during the game:
* Ghetto: Immigrants and Natives are in their own part of the space (dimension proportional to their respective numbers). These spaces cannot be invaded. They move randomly within their own space. As a consequence, the only contact between them occurs at the frontier of the space.densityIndex: This parameter is used to define the dimensions of the space on which players are positioned, which is proportional to sizeOfPopulation (the dimension is the square root of sizeOfPopulation multiplied by densityIndex), fixed at 2
* Random: Immigrants and Natives are dispersed randomly through the entire space, and move randomly. This option maximises the possibility of meetings between Immigrants and Natives.
* Invasion: This is a mixture of the previous two. Immigrants and Natives start in the Ghetto setting, and remain in it for a given number of rounds (timeInOwnSpace), after which they are allowed to "invade" each other's space (at a speed of invasionRate, which defines the number of "cells" by which the space of each player is increased). Players wait for some rounds (defined by timeBetweenInvasions) before increasing their space further. Eventually, they end up in the same situation as Random. We present the results for Random and Invasion with a speed of 3 cells.
endAfterRound: The game is based on an indefinitely repeated PD, and stops when an equilibrium is found (either all cooperate or all defect). However, it may happen that no equilibrium is ever found. This parameter allows stopping the simulation after a suitably large number of rounds in such cases, fixed at 100.
proportionIntelliImmigrants, proportionIntelliNatives,
proportionClever-Immigrants, proportionCleverNatives,
proportionNobelImmigrants, proportionNobelNatives:
These parameters define the proportion of different types of players in the population. These proportions are used respectively with the total number of Natives and Immigrants in the population. If the sum for Intelli, Clever and Nobel immigrants does not sum to 1, the remaining players are Normals. The same occurs for Natives.
For the purposes of the simulation, all players in the population were of the same intelligence (i.e. all Normal, all Intelli, all Clever or all Nobel). We plan to investigate mixed populations in further research.
payoffSetting: This parameter identifies the type of payoffs players get from the game. It takes so far four possible values ("first", "second", "third" and "fourth"). Each setting defines a different way to reward (or not) cooperation in the face of defection. In the simulations, the "second" payoff setting was used, in which cooperation is only rewarded when all other players in the group are cooperating.
pastTimes: For all players, this parameter defines their "memory", which can be interpreted as their attachment to their traditions. Typically, it has a higher value for Immigrants than for Natives (making the former more reluctant to change from non-cooperation to cooperation). In the simulation, pastTimes for the Natives was always fixed to 1. For the Immigrants, it was alternately fixed to 1 and 10.
learningRate: In addition to the memory parameter, players can learn at different speeds. In the simulations, all players had a learning rate of 1.
2"In early tales the golem was usually a perfect servant, his only fault being a too literal or mechanical fulfillment of his master's orders [...] It was the basis for Gustav Meyrink's novel Der Golem (1915) and for a classic of German silent films (1920)." (From "Golem", in the Encyclopaedia Britannica).
3Following Hume, Sugden (1986) suggests that the punishment of defectors tends to become an ethical value, thus reinforcing its viability as an enforcement devise. Falk et al. recently provided an experimental support to this hypothesis, to our opinion conclusive. "Our findings show that the violation of fairness principle is the most important driving force of sanctions, but, in addiction, a non-negligible part of sanctions is driven by spitefulness. We find surprisingly little evidence for strategic sanctions that are imposed to create future material benefits. While non-strategic sanctions are of major importance in our experiments, strategic sanctions seem to play a negligible role" (Falk, Fehr and Fischbacher, 2001).
4We impose the rule for players to change position in the space after each round in order to ensure that the groups are different in each round. In this case, players cannot get used to a given "neighbourhood" and are interacting with the entire population. In addition, the positioning of players in the space does not guarantee that they will all be able to play in any given round: some players may find themselves isolated and not have enough neighbours to form a group. Changing position at the end of a round also means that isolated players will get a chance to play in the following round of the simulation.
5This is the standard choice mechanism in game-theoretic economic literature. Other mechanisms may be plausible too. For instance, Eshel et al. (2000) and Uno and Namatame (1999) suppose that players observe neighbours, and imitate the successful ones. This pattern looks realistic, but the realism is easily lost if the imitation is simulated too simply. To our opinion both expectation and imitation deserve consideration. We hope to include this feature in further versions of our model.
6We would like to stress that there is no pejorative meaning in assuming that immigrants do not cooperate. This feature is made necessary by three logical steps. First, what we want to study is the robustness of cooperating conventions, so we must suppose one to be in force; second, the arrival of foreigners must be modelled, in our study, as the arrival of players adopting a different convention; third, it is much simpler to suppose, in a PD setting, that this convention imposes non-ccoperation.
7Each player occupies a position in the world, and can have at most 8 neighbours (each cell is surrounded by 8 other cells, which may or may not contain a player). During each round, each player tries to be part of a group of size n. This group is composed of his neighbours, and their neighbours, etc until it reaches the size n. Hence this grouping method is called "Chain of Neighbours" in the program, as all players in a group must touch at least another player by one side or one corner of the cell. Players participate in one group only in each round. Therefore, if a group cannot reach size n, its members stay inactive for the current round (we say that they "sleep" for a round).
8Note that this behaviour occurs in our model because punishment is present and significant. A smaller or zero punishment would teach Natives to defect, or lead to a mixed equilibrium.
9We do not include the Ghetto results as they are less informative. In the case of Normal players, the Immigrants learn to cooperate through interactions with Natives on the border of the Ghetto, but cooperation only diffuses throughout the Ghetto if the number of Immigrants is small (about 100 players, 5% of Immigrants). In the large population we consider, Normal players only learn to cooperate occasionally, if they happen to meet Natives in consecutive rounds. The "more intelligent" players (Intelli, Clever and Nobel) all evolve the same strategy: they learn that if Natives are in the group (i.e. they are playing on the border of the Ghetto), it is worth for them to cooperate, otherwise, they keep on defecting.
10As the "players" (immigrant and natives) are positioned randomly on the space at the beginning of the game, and move randomly afterwards, their observed behaviour depends also on the random seed in use at the beginning of the game. Running simulations changing the seed allows to see how much this behaviour depends on the randomness part of the game. In all cases, the variance observed was very small (of the order of 1 round).
11The program is available on request from the corresponding author.
AXELROD, R., (1986). An Evolutionary approach to norms. American Political Science Review, 80, 1095-1111.
BRETON, A. and M. BRETON, (1995). Nationalism revisited. In: A. Breton, G.L. Galeotti, P. Salmon and R. Wintrobe (eds.), Nationalism and Rationality. Cambridge: Cambridge University Press.
COOPER, B. and C. WALLACE, (2000). The Evolution of Partnership. Sociological Methods and Research, 28, 365-381.
ESHEL. I., D.K. HERREINER, L. SAMUELSON, E. SANSONE and FALK, A., E. FEHR and U. FISCHBACHER, (2001). Driving Forces of Informal sanctions. Institute for Empirical Research in Economics, University of Zurich, Working Paper 59.
HUME, D., (1740). A Treatise on Human Nature.
KIRCHKAMP, O., (2000). Spatial evolution of automata in the prisoners' dilemma. Journal of Economic Behavior and Organization 43, 239-262.
MAYNARD SMITH, J., (1982). Evolution and the theory of games. Cambridge: Cambridge University Press.
ORTONA, G., (2001). Economia del comportamento xenofobo. Torino: UTET.
SUGDEN, R., (1986). The Economics of Rights, Co-operation and Welfare. Oxford: Basil Blackwell.
SUGDEN, R., (1989). Spontaneous order. Journal of Economic Perspectives, 3, 85-97.
UNO, K. and A. NAMATAME, (1999). Evolving strategic behaviors through competitive interaction in the large. Paper presented to the 5th International Conference of the Society for Computational Economics, Boston, June 24-26.
VOGT, C., (2000). The evolution of cooperation in Prisoner's Dilemma with an endogenous learning mutant. Journal of Economic Behavior and Organization 42, 347-373.
WITT, U., (1986). Evolution and stability of cooperation without enforceable contracts. Kyklos, 39, 2, 245-266.
WITT, U., (1994), Moral norms and rationality within populations: an evolutionary theory. Paper presented to the annual meeting of the European Public Choice Society, Valencia.
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