Transcript Nash Equilibrium

Microeconomics 2
John Hey
Game theory (and a bit of bargaining theory)
• A homage to John Nash.
• Born 1928. Still alive (as far as Google knows).
• Spent some considerable time in
hospitals as a result of his ‘descent
into madness’.
• Won Nobel Prize for Economics in
1994 with John Harsanyi and
Reinhard Selten.
• He did lots of things, but in economics
he is associated with Game Theory
and Bargaining Theory.
Game theory
• Up to now we have considered situations in which
individuals take decisions independently of the decisions of
others.
• Today we consider situations of interdependence – games.
• It will be useful when we examine duopoly.
• Now widely used in economics for any kind of interactive
situation.
• We shall start with one-shot (static) simultaneous play
games without communication.
• Then we will look at finitely repeated, infinitely repeated and
randomly repeated games.
• We talk a little about sequential games and games with
communication (cheap talk).
Games
• In general, many players, many choices, many
repetitions.
• We start by considering (initially simultaneous play and
one-shot) games in which there are two players (1 and
2), each with two choices (A and B).
• The payoffs of both players depend on the choices of
both players.
• The concepts and ideas generalise to many players,
many choices and many repetitions.
• Indeed we will generalise to a continuum of choices (but
with just two players).
• The slides are colour-coded; Blue 1; Red 2.
•
(In this lecture we will not go to the Maple file.)
A Dominating Choice
• A player has a dominating choice if it is best
independently of the choice of the other player.
• (This concept is not to be confused with that of
a [Pareto] dominating outcome - which we
shall come across later.)
• In this first game, both players have a
dominating choice and thus prediction is easy.
• In the next game things are not so easy...
What might happen in this game?
• We see that Player 1 does not have a
dominating strategy (what is best for Player 1
depends on what Player 2 chooses).
• But Player 2 does: column A.
• Player 1 may be able to work this out, and
hence decide to play row A (which is his/her
best response to column A played by 2).
• We arrive at an equilibrium...
• ...which is an example of a Nash Equilibrium.
A Nash Equilibrium
• A combination of choices in a game is called a Nash
Equilibrium (NE) if neither player wants to change his or
her choice given the choice of the other player.
• This exists in Game 2: there it is unique.
• We might therefore want to ask:
• Does a Nash Equilibrium always exist/unique?
• We consider 3 more (types of game):
• Game 3: Two Nash Equilibria.
• Game 4: No equilibria in Pure Strategies (we introduce
the idea of a Mixed Strategy).
• Game 5: An implausible unique NE.
Two Nash Equilibria
• Note that here we have two Nash Equilibrium: (A,A) and
(B,B).
• Does the Nash theory say that one is more plausible than
other?
• No – simply that both exist.
• What, therefore, can we predict about behaviour?
• Nothing (especially in a one-shot game).
• Unless we add something to the Nash theory...
• ... perhaps relating to the out-of-equilibrium payoffs?
• Now a game without Nash equilibria (in pure strategies)...
A Mixed Strategy Nash Equilibrium
• Here we have what is called a Mixed Strategy NE.
• In this symmetrical example, each player should
play A and B with probabilities ½.
• Obviously to confuse the other player.
• What, therefore can we predict about behaviour?
• Nothing (especially in a one-shot game).
• Now let us look at the infamous Prisoner’s
Dilemma..
Pareto Dominance
• This refers to dominance in outcome rather than
in strategy.
• When one outcome is better for both players
than some other outcome, we say that the first
outcome Pareto Dominates the second.
• We note that the Nash Equilibrium (A,A) in the
Prisoners’ Dilemma is Pareto Dominated by
(B,B).
• We also note that this Nash Equilibrium is driven
by dominating choices for both players.
This is the Prisoner’s Dilemma
A
Player 1 B
Player 2
A
B
(5,5)
(99,0)
(0,99)
(96,96)
Another Prisoner’s Dilemma
A
Player 1 B
Player 2
A
B
(1,1)
(999,0)
(998,998)
(0,999)
And so on up – replacing ‘1’ by ever and ever smaller numbers (but bigger than
‘0’) and replacing ‘998’ by bigger and bigger numbers (but smaller than ‘999’) –
we get the same NE: (A,A).
And yet another Prisoner’s Dilemma
Player 2
A
Player 1 B
A
B
(1,1)
(99,-100)
(-100,99) (95,95)
And so on replacing ‘-100’ by smaller and smaller losses until it becomes ‘-100’
becomes ‘1’. Always the NE (driven by dominating strategies) is (A,A)
Summary so far
• A player has a dominating choice if this choice is best
independently of the choice of the other.
• A combination of choices in a game is called a Nash
equilibrium if neither player wants to change his or her
choice given the choice of the other player
• Games may have no Nash Equilibria (in pure strategies), a
unique Nash Equilibrium, or several Nash equilibria.
• There is no guarantee that Nash Equilibria are not Pareto
dominated by other outcomes.
• Notice crucially that the Nash theory does not say how an
equilibrium might be reached. It just says that an
equilibrium exists. Hence experiments...
Changing the rules
• Might some of these problems (in the one-shot game) go away
with communication?
• Cheap talk, no. Binding commitments, yes.
• Might some of these problems go away if the game is repeated?
• If we play it a finite and known number of times...
• ... No. (Backward induction shows this.)
• If we play it an infinite (impossible) or random number of times...
• ... Possibly.
• Might some of these problems go away if the game is
sequential?
• Not here, but possibly elsewhere.
Other solution concepts
• The Nash Equilibrium solution concept is
deeply embedded in economists’ minds.
• Notice that it leads to (A,A) in all variants
of the Prisoner’s Dilemma.
• What about Methodism?: “God first, others
second, you last”. “Do what is best for the
other person given what they are doing”.
• Or H*y*sm: “I assume everyone else is as
clever as me.”
A Continuum of Choices
• Usually Game Theory is applied in
situations with a finite number of choices,
but we can extend it to a continuum.
• When we consider duopoly, the two
players do not choose just from two
choices but choose the value of some
variable.
• We can use exactly the same concepts.
Nash bargaining theory
• Sequential, non-communicating, games may seem very
odd to you...
• ... in many situations we bargain with people.
• Wikipedia defines bargaining or haggling as “a type of
negotiation in which the buyer and seller of a good or
service dispute the price which will be paid and the exact
nature of the transaction that will take place, and
eventually come to an agreement.”
• Why does economics not have a theory of that?
• It does. Nash again. (and others)
• But odd again...
Nash’s solution to the bargaining problem
• Two players (X and Y) bargaining over the split of some
amount of money z between them. X gets x and Y gets
y. (x+y ≤ z). If they do not reach agreement each gets
zero.
• X (Y) has utility function u (v).(Unique only up to a linear
transformation.)
• Nash’s theory has solution: the maximisation of
[u(x)-u(0)][v(y)-v(0)]
subject to x+y=z.
• Again the theory does not say how (a sequence of offers
and acceptances/rejections?) this solution will be
achieved, just that it will/should. (Is this a normative or a positive theory?)
The axioms
(note the links)
• Invariant to affine transformations or Invariant to
equivalent utility representations. (Because EU utility
functions are unique only up to a linear transformation.)
• Pareto optimality (There is no other outcome which is better for both.)
• Independence of irrelevant alternatives (If A is preferred
to B out of the choice set {A,B}, then introducing a third alternative X, thus expanding
the choice set to {A,B,X}, must not make B preferable to A.
• Symmetry (We treat the two bargainers the same.)
•
Note what Nash says (Econometrica 1950): “… we assume that the two players are
highly rational, that each can accurately compare his desires for various things, that
they are equal in bargaining skill, and that each has full knowledge of the tastes and
preferences of the other.”
•
Experiments…
In conclusion
• Are the Nash equilibrium and the Nash
bargaining solution positive or normative?
• They both seem unrealistic as descriptions
of behaviour.
• Others think so too..
• ... and these thoughts have led to many
experiments looking at actual behaviour.
• But the theories stimulate thought!
Chapter 30
• Goodbye!