A strategy for achieving coordination amongst
agents in a minority game
Deepak Dhar
Indian Institute of Science Education and Research
Pune, INDIA
Topics in Applied Probability, TIFR, April 1, 2017
Deepak Dhar
Minority games
Work done with
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Hardik Rajpal ( I.I.T. Kharagpur)
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Outline
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The Minority Game for agents with long future-horizon
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The Periodic state for optimizing the long-time average payoff
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The Strategy
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Average time to reach coordination
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Log-periodic oscillations in the average time
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Summary
Deepak Dhar
Minority games
Minority Games
El Farol Bar problem [Brian Arthur (1994)]
Minority game [ Challet and Zhang (1997)]
Minority game is a prototypical model of Econophysics.
Any ”winning strategy” will fail, if followed by all.
Simple nontrivial model of interacting agents showing
self-organization, learning, adaptation, co-evolution
Deepak Dhar
Minority games
Definition of the game
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Repeated game involving 2M + 1 agents, two restaurants
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Payoff 1, if in the minority; else 0.
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No direct communication between agents. Only past history
of total attendances known to all.
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All agents are Rational, and Equal, want to maximize their
personal long-time average payoff per day
Deepak Dhar
Minority games
Optimality of the periodic state
Clearly, there are at most M winners per day, and so the long-time
average payoff per person per day satisfies the inequality
P≤
M
.
2M + 1
This payoff is attainable, if the agents can get into a periodic state
of period (2M + 1).
Many possible periodic states.
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Minority games
The possibility of coalitions
It is possible that some agents can get payoff better that P, while
others get < P, if they can form a coalition.
Consider M = 1. Three agents X, Y and Z.
If X and Y together reach an agreement that X always goes to A,
and Y always goes to B, then one of them always wins, and Z
never.
Then, average payoff per day for X or Y is 1/2, and 0 for Z.
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Coalitions? (cont.)
The agents can reach this coalition state, without any direct
communication.
For example, playing randomly, X and Y may notice that they have
won more often when X goes to A, and Y goes to B, and then
stick to this winning strategy.
In reaching this coalition state, X does not know who is the
coalition partner.
Also, the coalition strategy may be more complicated: e.g. X
chooses periodic repetition of AABAB, and Y of BBABA.
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Minority games
If Z finds herself out of the coalition, what can she do?
If she could know the period of strategies of X and Y, she could
also adopt a periodic choice of same period.
Then, the long-time average payoff of X or Y may become less
than 1/3, and she will have no incentive to stay in the coalition.
Unfortunately, this knowledge is not available to Z.
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Minority games
Will selfish agents try to form a coalition, and get more than
others, or be satisfied with a periodic state, that is egalitarian?
Clearly, there is probability 1/3 that even if a coalition is formed,
the person will be out of it!
We study the case of Rational agents, who want to maximize their
personal expected payoff per day, and hence prefer the periodic
state.
Deepak Dhar
Minority games
If all agents want to reach a periodic state, but can not talk to
each other directly, how can this be done?
This then becomes a Coordination game.
The Strategy:
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The strategy assigns to each agent an integer ( called tag),
which is known to the agent.
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Once all agents are assigned tags, they follow a predecided
periodic cycle.
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Our algorithm works in two stages.
In stage I, Exactly M of the (2M + 1) agents are assigned tag 0.
In stage II, the remaining (M + 1) agents are assigned tags from 1
to M + 1.
Once all agents know their tags, all agents adhere to a predecided
periodic pattern of choices.
e.g. All agents with tag 0 always stay in Restaurant A. Agent with
tag i stays in restaurant B, on all days except on day
(2i − 1) (mod 2M + 1), when he shifts to A.
Each agent wins at least r times in any consecutive 2r + 1 days.
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Stage I:
Initially, each agent chooses the restaurant to go to randomly with
equal probability.
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If at the end of day, the number of people is exactly M in the
minority restaurant, each is assigned a tag 0, and stage ends.
Else, if the break is (M − ∆, M + ∆ + 1) with (∆ > 0),
those in the minority stay put; those the majority shift with
probability ∆/(M + ∆ + 1). Repeat.
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Minority games
Stage II:
This is a recursive algorithm.
At the beginning, there are an identified set S of people, who are
to be assigned tag, all in the same restaurant.
Each agent knows if he is in this set or not.
If |S| = 1, the single person is assigned the available tag, and the
algorithm ends.
Else
Each agent in S shifts with prob 1/2. Other agents stay put.
At the end of day, it is known how many agents jumped. Thus, S
is divided into two sets: The smaller set S1 , and the larger set S2 .
If of equal size, those who jumped are called S1 . Now recursively
assign tags, first to S1 , and then to S2 .
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Average time to assign tags
Let the Average time in the second stage to a set of n agents be
Tn . Clearly,
T0 = T1 = 0; T2 = 2.
The probability that exactly r people shift is nr 2−n . Then, Tn
satisfies the linear equation
Tn = 1 +
n
X
Prob(r )[Tr + Tn−r ], for n ≥ 2.
r =0
We can thus determine all the Tn recursively. For example,
T3 = 10/3, and T4 = 100/21.
The values of Tn , for n ≤ 30 were determined numerically are
shown in Fig. 1. We see that Tn increases approximately as
1.4449n.
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Figure: Numerically determined exact values of Tn for n ≤ 30. The
equation of the approximate linear fit here is y = 1.4449x − 1.0451.
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This can also be determined analytically.
Define the generating function
T (x) =
∞
X
Tr x r ,
r =1
we find that T (x) satisfies the equation
T (x) =
x2
4
x
+
T(
)
(1 − x/2) (2 − x) 2 − x
This equation can be converted to simpler equation using
H(y ), we get
y = 1/(1 + x), and T (x) = 1+y
y2
H(y ) =
1+y
+ H(2y ),
y2
which has the solution
H(y ) =
∞
X
s=0
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y 2s
.
+ 1)2
(2s y
Minority games
For small y , the summation over s may be extended to −∞.
Then, we get a periodic function in log2 y . Equivalently,
Tn ∼ n [A + B cos (2π log2 n) + ..]
8e-11
6e-11
4e-11
2e-11
0
-2e-11
-4e-11
-6e-11
-8e-11
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure: Log-periodic oscillations in the function H(y ) as a function of
log2 y , determined numerically, about the mean value 1.44269504089.
Note the small amplitude of the oscillations.
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Minority games
Summary and Concluding remarks:
We described an algorithm for agents in a minority to generate a
cyclic state without direct communication.
The time varies linearly with n, but the coefficient shows a small
log-periodic oscillation with n.
No simple qualitative argument to see why the amplitude of
oscillations is so small O(10−10 ).
In most earlier known instances, log-periodic oscillations occur in
the sub-leading term. Here, in the leading term.
Deepak Dhar
Minority games
Thank You
Deepak Dhar
Minority games