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GAME THEORY:
WEEK 10
Problem set
Problems. Please complete the following problems.
From Ben Polak.
Consider the following game involving two real players and a chance
move by ‘nature’. America and Russia have the nuclear capability to
destroy each other. ‘Nature’ tosses a fair coin so that with probability
0.5 America moves first and Russia moves second, and with probability
0.5 Russia moves first and America moves second.
For now, assume that both countries observe nature’s choice so they
know whether they are first or second. The country who moves first
decides whether to ‘fire’ its missiles or to ‘wait’. If it fires then the
game ends: the country who fired gets a payoff of −1 and the other
country gets −4. If the first country waits then the second country
gets to move. It too must decide whether to ‘fire’ or ‘wait’. If it fires
then the game ends and it gets −1 and the other country gets −4.
If it ‘waits’ then both countries get 0. Assume each country seeks to
maximize its expected payoff.
Treat this as one game rather than two different games. The game
tree is shown below. The first payoff refers to America and the second
to Russia. There are no payoffs to nature.
N
0.5
0.5
Am
Ru
f
f
w
Ru
−1, −4
f
−4, −1
w
Am
−4, −1
w
f
−1, −4
0, 0
w
0, 0
(1) What makes this a game of perfect information? Write down
the definition of a strategy in an extensive form game, and identify the possible strategies for America and for Russia in this
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GAME THEORY: WEEK 10
game.
Check the notes and Osborne. This is just restating the definition of an extensive form game and labelling the appropriate
parts.
(2) Find and explain any pure-strategy subgame-perfect equilibria
(SPE), making clear what constitutes a subgame. Are there
any Nash equilibria which are not SPE?
To get the SPE, use backward induction. At the end of the
game, nobody would choose to fire. As such, in the penultimate
node, nobody chooses to fire. Thus, nobody chooses to fire (everybody waits) is the SPE.
(3) Now suppose that neither Russia or America observes the move
by nature, or each other’s move. That is, should a country be
called upon to move, it does not know whether it is the first
mover or whether it is the second mover and the other country
chose ‘wait’. Again, treat this as one game. Draw a game tree
similar to the figure but for this new game. Indicate clearly
which nodes are in the same information sets.
N
0.5
0.5
Am
Ru
f
f
w
Ru
−1, −4
f
−4, −1
w
0, 0
w
Am
−4, −1
f
−1, −4
w
0, 0
(4) Identify the possible strategies for America and for Russia in
the game from part (3). Find and explain carefully two ‘symmetric’ pure-strategy SPEs in this game that have very different
outcomes.
Effectively each country now has only two possible actions:
fire or wait. Unlike in the previous example, there is no clear
‘end node’ in this game for either player. They believe with 50%
that they are the first mover and 50% that they are the second
mover. If they fire, they will get a payoff of -1 with certainty
(either they are the first mover and this ends the game or they
GAME THEORY:
WEEK 10
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are the second mover and this ends the game). However, if they
choose to wait this could end the game (50% chance) or it could
give Russia a chance to move. Suppose they believe Russia will
choose to wait with probability 1 − p and fire with probability p.
In that case, their payoff to waiting is −2p. If America believes
that p=0 then they will wait (and symmetrically for Russia) so
this is, in a sense, self-fulfilling. If, however, America believes
there is a chance p > 50% that Russia will fire then America
will preemptively fire on them! This is also self-fulfilling for
Russia.
So there is one equilibrium where both parties believe the other
would wait (low p) and consequently they wait. There is another
equilibrium where they believe their opponent will fire (high p)
and both parties then fire.
(5) Now suppose that America can observe the move by nature and
also (when it is the second mover) Russia’s move. Russia knows
what America can observe, but, as before, Russia can observe
neither nature’s nor America’s move. Draw the game tree for
this game. Argue whether you think the world is a safer place
or a more dangerous place now that America is better informed
than Russia. That is, compare the SPE of this game with the
SPE of the game of parts (2) and (4).
N
0.5
0.5
Am
Ru
f
f
w
Ru
−1, −4
f
−4, −1
w
0, 0
w
Am
−4, −1
f
−1, −4
w
0, 0
In this game, when America moves second, she will choose to
wait (this is SPE). Russia knows this and so there must be a
probability that America waits of at least 50% – in fact, if Russia moves second and chooses to wait then the game ends! So
Russia should conclude that by waiting (even though he doesn’t
know if America gets to move again) they will get the best possible payoff. This then should induce Russia to also wait by
the reasoning in part (4). This informational should make the
world safer.