=Paper=
{{Paper
|id=None
|storemode=property
|title=Simulation of the Expanded Iterated Version of the Prisoner's Dilemma Game
|pdfUrl=https://ceur-ws.org/Vol-716/ICTERI-2011-CEUR-WS-paper-5-p-82-86.pdf
|volume=Vol-716
|dblpUrl=https://dblp.org/rec/conf/icteri/VlasovaVS11
}}
==Simulation of the Expanded Iterated Version of the Prisoner's Dilemma Game==
https://ceur-ws.org/Vol-716/ICTERI-2011-CEUR-WS-paper-5-p-82-86.pdf
Simulation of the Enhanced Version of Prisoner’s
Dilemma Game
Tatyana Vlasova, Maryna Vladymyrova and Dmitry Shabanov
V.N. Karazin Kharkiv National University, 4, Svobody Sqr., Kharkiv, 61077, Ukraine
vlasova.tatyana@bk.ru, vladymyrova@gmail.com, d.a.shabanov@gmail.com
Abstract. This paper presents the model and software to explore pair
interactions of objects with different behavior and their patterns. The research
is based on the enhanced version of a classic prisoner’s dilemma game. The
non-cooperative finite and infinite pair games with non-zero sums are
investigated. Pure and mixed strategies with finite and infinite memory
developed by Biology School of V. N. Karazin Kharkiv National University are
used to analyze the results.
1 Introduction
This paper presents the model and software to explore pair interactions of biological
objects with specified behavior. The research was carried out on request of the
herpetology department of V. N. Karazin Kharkiv National University and its results
are used in educational process to illustrate some topics of the ecology discipline as
well as in biology students’ research for studying patterns of pair behavior of some
biological species. Our model is based on the classic prisoner’s dilemma game [1,2,3]
as it is widely accepted as the model to study the pair interactions and behavior of
different agents from interactions of animals in nature up to economical transactions
in human world [4, 5, 6, 7, 8]. The main goal of the game in its classic version is to
get maximal score doing preset number of steps with given values of the fine, the
cooperation bonus and the cooperation award. Each participant can remember not
more than two its own and the opponent’s previous actions.
In distinction to classic case we enhance the rules of the game by allowing more
complex behavior of agents depending on their ability to remember their own or
opponent’s previous steps and on values of fines, bonuses and awards.
The important factor which influences the evolution of living organisms including
humans is their communications with environment. It is only typical for such
communications to have conflicts of interest between opponents, absence of
information about future actions of an opponent and need to foresee its future actions
only on the base of the prehistory of similar interactions. The paradox of the game
clearly shows the contradiction between group interests and individual ones: what is
optimal for the group of two is not good for each member of the group. In fact the
same is true for multilevel systems with optimization functions on different levels:
their behavior is intuitively unpredictable and even paradoxical.
Simulation of the Enhanced Version of Prisoner's Dilemma Game 83
It is absolutely obvious that the outcome of the game fully depends on the
participants’ strategies. Here we define strategy in a slightly different way than it is
done in the game theory.
Since the objective of this research is to explore the pair interactions in real
biological environment it is natural to define strategy as a set of rules used by a
participant to make its next step. Because of the infinite variety of strategies for
different biological objects we use simulation with strategies constructed by the
experimenter. The developed software allows experimenter to set different strategies
as input information for simulation. The main goal of experimenter is to find optimal
strategy and the value of the game in their classical sense.
2 Model and simulation description
We consider the pair non-cooperative finite and infinite games with a non-zero sum
[11; 12; 14] so the general result can be both positive (in the case when participants
cooperate during the whole game) and negative. We take the game in its normal form
N N ; X 1 ,..., X i ,..., X n ; K1 ,..., K i ,..., K n , where N is the notion of the game,
N {1,..., i,..., n} is a set of participants, X i {xi } is a set of strategies for the i-th
participant and K i ( x1 ,..., xi ,..., xn ) is the gain function for the i-th participant, the
value of which is the gain obtained by the i-th participant if participants use
strategies ( x1 ,..., xi ,..., xn ) . In our model we allow participants to use pure or mixed
strategies. Pure strategy is just definite sequence of steps i. e. it can be represented by
any element xi X i . Mixed strategy can be a simple set of pure strategies or a set of
pure strategies with given probabilities distribution.
Any strategy as a set of rules depends on the participant’s memory depth or its type
of memory. Here we suppose that each participant remembers its previous actions or
the previous actions of the opponent. If the participant has zero memory depth it can
make predefined actions during the whole game or make spontaneous actions. If the
participant has finite memory depth it uses pure strategy depending on the actions of
its opponent. If the participant has infinite memory depth only the absolute value of
its current gain influences its actions.
The pure strategy is a response of the participant to the actions of its opponent. We
consider the following cases:
No response (leads to spontaneous actions)
The response to the definite set of actions
The response to the small value of the own gain or to the big value of the opponent’s
gain
The response to the big value of the own losses or to the small value of the opponent’s
losses
In this case we achieve the goal of maximization of the participant’s own gain.
Similarly when the participant remembers only its opponent’s history of actions
and responds to them we achieve the goal of minimization of the opponent’s gain.
We use finite automation with two states which are “accept” and “reject” to model
the game [8, 12]. The input information for the automation is as follows: participants’
84 T. Vlasova, M. Vladymyrova and D. Shabanov
strategies, values of the gain function, transition rules for a participant’s behavior, and
initial conditions. Transition rules depend on the type of a participant’s strategy which
can be pure or mixed. Simulation repeats the preset number of times or till the
winning of one of its participants.
There exist four scenarios of simulation each corresponding to four different goals.
First scenario is the experiment between the two chosen strategies (“pure-pure”,
“mixed-pure”, “mixed-mixed”), when the values of gain/bonus/fine (M, L, and K,
respectively) are fixed. The simulation results are presented by the graph, where x-
axis denotes the sequence of steps and y-axis denotes the quantitative characteristics
of the gain/loss (Fig.1).
Fig. 1. Simulation results with two strategies (one is random).
The result data of this scenario show that among the strategies with zero and finite
memory, the strategy of random steps and zero memory wins in most cases. Among
the strategies with finite and infinite memory the finite memory strategy wins in most
cases. Analogous conclusions were made previously by R. Dockins [6] when he
analyzed the classical version of the game.
Analyzing the obtained results of pair interaction between the pure and mixed
strategies we may conclude that the mixed strategy is more advantageous than any of
pure strategies. This can be explained by the flexible behavior of the participant with
the mixed strategy.
The second scenario is the evaluation of competition between the one fixed
strategy and the variety of others. The results of the experiment are presented by the
bar chart displaying the number of win points over each of the chosen strategies.
Besides the graphical visualization one can look through the steps history (absolute
values of gains or losses) of each of the simulation participants.
The third scenario allows experimenter to approve or disprove the hypothesis that
the strategy of a participant depends on the gain function. In order to do it we
implemented the feature which allows experimenter to conduct the simulation with
fluctuating parameters M, L, K. In this case the results are presented by the graph and
Simulation of the Enhanced Version of Prisoner's Dilemma Game 85
the table showing the data of each participant’s results for chosen strategies with
respect to varying values of K, L, and M (Fig.2).
Fig. 2. Simulation results with fluctuating values of K, L, and M.
The results of the experiments show that the outcome of the game depends on the
values of the gain function. So one can find experimentally the values of the gain
function for any given strategy to be optimal.
Fourth scenario deals with a strategy as an element of the given set. We do not use
the formal methods of working with such sets but accepted the way used by
experimenters (biologists) to form them. In this case our goal was to find the
probability for a strategy from one set to win a strategy from another one. For
example our experimenters divided strategies into three sets namely provocative,
forgiving and neither provocative nor forgiving exactly in the same way as it is done
in R. Axelrod’s experiment [4]. In our case a provocative strategy means immediate
change of behavior in condition of the participant’s own loss (or the opponent’s gain)
and keeping it till the next loss or till the end of the game. A forgiving strategy means
that the participant changes its behavior under the same conditions but keeps it only
some limited time (definite number of steps). In some way one can see it as follows:
in forgiving strategy the participant only fights back as a response to the smack while
in provocative strategy the participant not only fights back as a response to the smack
but retaliates. Such division is just conditional as it reflects the view of experimenters.
The results of experiments show that the provocative strategies win in more cases
than forgiving ones.
86 T. Vlasova, M. Vladymyrova and D. Shabanov
3 Conclusion
The software for simulation the enhanced version of the prisoner’s dilemma game to
set up experiments and explore pair interactions of objects with different behavior has
been developed. The software allows experimenter to set pure and mixed strategies
with finite and infinite memory. The simulation depends on its goal: whether it is
maximization of the participant’s gain or minimization of the opponent’s loss. The
participant’s gain depends on the values of the gain function. If the participants’
strategies are fixed then one can find experimentally such values of the gain / bonus /
fine that the strategy of one of the participants is optimal. The results of the
simulation comply with those given in literature in the case of classic game which can
be accepted as an adequacy of the model.
The software was tested and operates at Biology School of V. N. Karazin Kharkiv
National University but it can be successfully used at other schools and fields such as
economy, sociology, etc.
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(1981) 1390-1396
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http://www.prisoners-dilemma.com/
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