Econ 414, Daniel R. Vincent
Introduction to Game
Theory III
Common Knowledge and
Payoffs
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Outline of Lecture
a1. The Assumption of Common
Knowledge
` Its implication for games
` A puzzle.
a2. Payoffs
`Uncertainty and Probability
`Expected utility.
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Econ 414, Daniel R. Vincent
1. Common Knowledge
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Common Knowledge
aWe say that something is “Common Knowledge”
if “He knows that she knows that he knows that
she knows .... “ ad infinitum. Everyone knows it
and everyone knows everyone knows it etc.
aIn game theory, we commonly assume that
`All the potential players are CK,
`The strategies are CK.
`The payoffs are CK.
aSo everyone knows “the details of the game”.
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Econ 414, Daniel R. Vincent
Common Knowledge
aIs the assumption of Common Knowledge so
strong that it rules out most interesting strategic
situations?
aIn particular, does it rule out any situation
where players have some uncertainty?
aNot necessarily, all we need is that, at some
level, the structure of the game is commonly
known.
aThe next example illustrates this feature.
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Example of no CK of a
payoff
a Here the payoff off player 2 is not fully
known.
1
2
(3,x)
(0,2)
(0,1)
(2,5)
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Econ 414, Daniel R. Vincent
Example: Continued
aPlayer 1 cannot determine an optimal strategy
without knowing what Player 2’s optimal
strategy is.
aEven if Player 2 knows x if player 1 does not, he
cannot analyze the game well enough to play
sensibly.
aWe can fix this problem without having to give
Player 1 too much information…
aWe add an articifical stage where the
uncertainty is generated by “Nature”.
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Example of no CK of a
payoff: A potential cure?
N
a
.5
.5
1
2
(3,0)
(0,2)
(0,1)
(2,5)
(3,3)
(0,2)
(0,1)
(2,5)
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Econ 414, Daniel R. Vincent
An example of No CK
aHere the “fix” is to assume that the players are
in a bigger game, where a random draw is
made first determining x.
aThe details of the larger game (including the
possibility of nature’s random move) is
Commonly Known.
aNature’s actual choice is not known to player 1
but player 1 will know the probabilities used for
the draw.
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An Example Where
Everyone Knowing
Everything is Not Common
Knowledge.
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Econ 414, Daniel R. Vincent
The Island People and
their Dots
a There is an island of preindustrial civilization. The society is made
up of five very sensitive people. They meet each morning at the
island’s best spring for water. If they do not use this spring, they
have to drink the awful water on the other side of the island.
a Each person on this island has a fifty percent chance of being born
with a blue dot on the forehead. They are all very sensitive about
this dot and if they knew they had a dot, they would not be seen in
public and would not use the fountain.
a Unfortunately, fate would have it that all five people on the island
were born with the dot. However, fortunately, there are no mirrors
on the island and so they continue to meet every morning at the
spring.
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The Island People and
their Dots
a One day, Blabbermouth Bob, a visiting anthropologist,
happens to say “At least one of you has a blue dot on
your forehead.”
a Has anything of significance occurred? What will
happen?
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Econ 414, Daniel R. Vincent
The Island People and
their Dots
a Analysis: Although every person on the island knows the
information that Bob has broadcast, it is not common
knowledge. That is because, no one can rule out the
conjecture that A believes that B believes that A believes
that B believes that A believes that there are no blue
dots.
a Bob’s announcement has ruled this possibility out. What
are the possible events? 0 blue dots, 1 blue dot,...,5
blue dots. Bob has ruled out only the first, but he has
done it in a way that makes it common knowledge.
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The Island People and
their Dots
a If case 1 dot had occurred, then when the
announcement is made, the one person with the dot
would know it immediately and leave. Since this does
not happen, the group meets the next day and it would
be common knowledge that only cases 2,3,4, and 5
remain.
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Econ 414, Daniel R. Vincent
The Island People and
their Dots
a If case 2 had occurred, then two of the group would see
only one dot, infer that they also have a dot and leave
immediately. This does not happen so the group meets
a third day with cases 3,4 and 5 remaining. Now, if it
was case 3, three members would see only two dots
and would leave. This does not happen so the fourth
day they all come back. On this day, four members
would see three dots under case 4, would infer they
have the fourth and would leave. Since this does not
happen, there is only one case left and on the fifth day
no one returns.
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2. Payoffs
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Econ 414, Daniel R. Vincent
Payoffs
aRecall that one of the three components
of a game is a full description of the
consequences of every possible play of
the game for every player.
aThat is, we have to describe how players
“care” about the outcomes of a game.
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Payoffs
a Payoffs must capture all aspects of the
outcomes of a game that a player may
care about.
aWe assume that a player is “rational” that
is, the player acts to maximize her own
expected payoff
aDoes this imply we must presume players
are selfish?
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Econ 414, Daniel R. Vincent
Example of the selfish
father? Incorrect version?
a
Father
Buy the car
No Car
Father
Drives her
(3,3)
Friends
Drive
(0,2)
Daughter
Careless
Careful
(0,1)
(2,5)
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The Selfish Father
aWe typically expect that parents do not care
ONLY about themselves but also their offspring.
aDoes this keep us from analyzing games in
these situations?
aNot necessarily.
aIf we wanted to argue that of course the father
will prefer his daughter to do well, then we need
to incorporate that fact in the payoffs.
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Econ 414, Daniel R. Vincent
Example of the selfish
father? Correct Version?
a
Father
Daughter
(3,3)
(0,2)
(0,1)
(5,5)
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Selfish Players
aThe main point is that players can care
about each other’s welfare.
aIf we want to capture that
interdependence, though, we need to be
sure to do it in the specification of
payoffs.
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Econ 414, Daniel R. Vincent
More on Payoffs
aIn the next lecture, we develop some
further tools to help us incorporate
payoffs when there is uncertainty in the
game.
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