This is a Game Theory question on Evolutionary Strategies: Consider TFT (tit for
ID: 3123257 • Letter: T
Question
This is a Game Theory question on Evolutionary Strategies:
Consider TFT (tit for tat): cooperate in the first interaction and then cooperate if the other player cooperated on the previous interaction and defect if he defected, TF2T (as defined above) and a population where 30% of players play TF2T, 30% play TFT and 40% play a strategy s. (1) Can you construct a strategy s for which u(s) > u(TF2T) > u(TFT)? If you can, please provide a (careful) definition of s. If you think such strategy does not exist, explain why. (2) Can you construct a strategy s for which u(TF2T) > u(TFT) > u(s)? If you can, please provide a (careful) definition of s. If you think such strategy does not exist, explain why.Explanation / Answer
Evolutionary games
The bandwidth choice game can be given a different interpretation where it applies to a
large
population
of identical players. Equilibrium can then be viewed as the outcome of
a
dynamic process
rather than of conscious rational analysis.
@
@
@
5
5
1
0
0
1
1
1
I
II
High
Low
High
Low
Figure 5. The bandwidth choice game.
15
Figure 5 shows the bandwidth choice game where each player has the two strategies
High
and
Low
. The positive payoff of 5 for each player for the strategy combination
(
High, High
) makes this an even more preferable equilibrium than in the case discussed
above.
In the evolutionary interpretation, there is a large population of individuals, each of
which can adopt one of the strategies. The game describes the payoffs that result when
two of these individuals meet. The dynamics of this game are based on assuming that
each strategy is played by a certain
fraction
of individuals. Then, given this distribution
of strategies, individuals with better
average payoff
will be more successful than others,
so that their proportion in the population increases over time. This, in turn, may affect
which strategies are better than others. In many cases, in particular in symmetric games
with only two possible strategies, the dynamic process will move to an equilibrium.
In the example of Figure 5, a certain fraction of users connected to a network will
already have
High
or
Low
bandwidth equipment. For example, suppose that one quarter
of the users has chosen
High
and three quarters have chosen
Low
. It is useful to assign
these as percentages to the columns, which represent the strategies of player II. A new
user, as player I, is then to decide between
High
and
Low
, where his payoff depends on the
given fractions. Here it will be
1
/
4
×
5+3
/
4
×
0 = 1
.
25
when player I chooses
High
, and
1
/
4
×
1 + 3
/
4
×
1 = 1
when player I chooses
Low
. Given the average payoff that player I
can expect when interacting with other users, player I will be better off by choosing
High
,
and so decides on that strategy. Then, player I joins the population as a
High
user. The
proportion of individuals of type
High
therefore increases, and over time the advantage
of that strategy will become even more pronounced. In addition, users replacing their
equipment will make the same calculation, and therefore also switch from
Low
to
High
.
Eventually, everyone plays
High
as the only surviving strategy, which corresponds to the
equilibrium in the top left cell in Figure 5.
The long-term outcome where only high-bandwidth equipment is selected depends on
there being an initial fraction of high-bandwidth users that is large enough. For example, if
only ten percent have chosen
High
, then the expected payoff for
High
is
0
.
1
×
5+0
.
9
×
0 =
0
.
5
which is less than the expected payoff 1 for
Low
(which is always 1, irrespective of
the distribution of users in the population). Then, by the same logic as before, the fraction
of
Low
users increases, moving to the bottom right cell of the game as the equilibrium. It
16
is easy to see that the critical fraction of
High
users so that this will take off as the better
strategy is 1/5. (When new technology makes high-bandwidth equipment cheaper, this
increases the payoff 0 to the
High
user who is meeting
Low
, which changes the game.)
The evolutionary, population-dynamic view of games is useful because it does not
require the assumption that all players are sophisticated and think the others are also ra-
tional, which is often unrealistic. Instead, the notion of rationality is replaced with the
much weaker concept of
reproductive success
Consider the following strategy “Tit for Two Tats” (TF2T): Cooperate in periods 1 and 2. Thereafter defect in any period k>2 if and only if your opponent defected in k-1 and k-2.
(a) Consider best response strategies to TF2T in a discounted repeated game with sufficiently close to zero.
Is it possible to construct two strategies, j and j*, such that both of them are best responses to TF2T and (TF2T, j) is in Nash equilibrium while (TF2T, j*) is not?
YES NO (circle one)
If your answer is YES then give an example of two strategies like that. Whenever possible use the strategies defined in the lecture notes, otherwise define a strategy of your own.
j = ……………..
j* = ……………..
(b) Consider best response strategies to TF2T in a discounted repeated game with sufficiently close to one.
Is it possible to construct two strategies, j and j*, such that both of them are best responses to TF2T and (TF2T, j) is in Nash equilibrium while (TF2T, j*) is not?
YES NO (circle one)
please go trough this link it may help you to solve your problem:
https://gradschoolpapers.com/2017/04/12/gametheory/
https://www.cs.ubc.ca/~kevinlb/teaching/isci330%20-%202006-7/Lectures/lect18.pdf
http://www.cdam.lse.ac.uk/Reports/Files/cdam-2001-09.pdf
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