Title: Game Theory Iterated Dominance, Rationalizability
1Game Theory Iterated Dominance, Rationalizability
- Univ. Prof.dr. M.C.W. Janssen
- University of Vienna
- Winter semester 2008-9
- Week 42 (October 13, 15)
2Dominated Strategies
- A strategy is strictly dominated if there is
another (mixed) strategy that is always better,
no matter what the opponents are choosing - Formally, strategy sets for player i are denoted
by Si and Si (for mixed strategies) pay-offs by
ui(si ,s-i ) - Strategy si ? Si is dominated if there exists
another strategy si ? Si s.t. - ui(si ,s-i )lt ui(si ,s-i ) for all s-i ?S-i
3Example I
- Why do we explicitly allow for domination by
mixed strategies? - M is not dominated by a pure strategy, but it is
strictly dominated by a mixed strategy, e.g., by
(½,0,½) - Thus, M is strictly dominated
4Clarification II
- Why do we not explicitly allow for other players
playing mixed strategies? - If a strategy is dominated for all pure
strategies of the other players, then it is also
dominated for all mixed strategies - Vice versa, if a strategy is dominated for all
mixed strategies of the other players, then it is
also dominated for all pure strategies (trivial)
5Example II mixed strategies can also be
dominated
- Clearly, a mixed strategy given positive weight
to a dominated pure strategy is itself dominated - But, other mixed strategies can be strictly
dominated as well - None of the pure strategies is dominated here,
but the mixed strategy (½,½,0) is strictly
dominated by D
6Example III Second price sealed-bid auction
- Suppose there are I bidders, each with valuation
vi for the object. - Suppose here valuation is known.
- Sealed bid implies it is a simultaneous move
game, si bi second price means highest bidder
wins and pays second-highest bid - Bidding your valuation is dominant strategy,
i.e., all other strategies are dominated by si
vi - Proof
7Iterated dominance
- Let S0i Si and S0i Si and define iteratively
all strategies that are not dominated yet as Sni
si ? Sn-1i ? there is no si ? Sn-1i
s.t. - ui(si ,s-i ) gt ui(si ,s-i ) for all s-i ?Sn-1-i
- S?i ?n Sni
- S?i is set of all mixed strategies si such that
there does not exists a si with ui(si ,s-i ) gt
ui(si ,s-i ) for all s-i ? S?-i - The game is dominance solvable if for all players
S?i and thus contains only one element - If game is finite, a finite number of iterations
will suffice - Justification in terms of common knowledge of
rationality
8(Iterated) Weak dominance (IEWDS)
- Same procedure can be applied to weak dominance
- Formally, strategy si ? Si is weakly dominated
if there exists another strategy si ? Si s.t. - ui(si ,s-i ) ui(si ,s-i ) for all s-i ?S-i
with strict inequality for some s-i
9(Iterated) Weak dominance (IEWDS)
- Order of elimination may matter for final outcome
- E.g., one may eliminate in this order U (weakly
dominated by D), then L. Outcome M and D for
player 1 and R for player 2 - Or, eliminate in this order M, R and then U.
Outcome (D,L)
10Justification for IEDS clear
- Order of elimination for iterated dominance does
not matter - Main idea if some strategy is strictly dominated
for a larger set of strategies of the opponent,
then it is also dominated for a smaller set.
Hence, if one does not immediately delete a
strategy, it should be deleted at some point. - Also, one does not need to delete immediately
mixed strategies
11Application to Cournot oligopoly
- N firms
- Linear market demand P(Q) a-bQ
- Linear cost C(q) cq
- Firms want to max (P(qiQ-i) qi C(qi)
12Best response functions for 2 firms
- What can be eliminated in first round?
- everything that is a best response cannot be
eliminated. - only possibility is quantities above monopoly
output - fact that they are not best reponses does not
necessarily imply they are dominated
qM
13Elimination in rounds I
- Round 1 All outputs above monopoly level are
dominated - ?p/?q a b(n-1)q-i c 2bqi lt 0 for all q-i
and qi gt (a c)/2b - thus, monopoly output gives always higher profit
than larger output levels - Round 2 for n2, all output levels smaller than
(a c)/4b ( optimal response to monopoly
output) are dominated - ?p/?q a bq-i c 2bqi gt 0 for all q-i lt (a
c)/2b and qi lt (a c)/4b
14Elimination in rounds II
- Round 2 for ngt2, no output levels are dominated
- ?p/?q a b(n-1)q-i c 2bqi lt 0 for q-i (a
c)/2b and qi gt 0 - Hence, there are strategies for the other players
that are not dominated such that qi 0 is a
better response than qi gt 0. - Round 3 (and more) for n gt 2, nothing changes
- Round 3 (for n 2), output levels larger than
3(a c)/8b are dominated by 3(a c)/8b - ?p/?q a bq-i c 2bqi lt 0 for q-i gt (a
c)/4b and qi gt 3(a c)/8b
15Elimination in rounds III
- Round 4, 6, 8, for n2, again low output
levels are dominated - Suppose that q,q are strategies that remain
after previous round, with q lt qc lt q lt qm and
q is best response to q - Thus, q (a-c)/2b - q/2 and best response to q
is (a-c)/2b q/2 (a-c)/4b q/4 gt q if, and
only if, q lt (a-c)/3b qc - For any q lt qc the following is true ?p/?q a
bq-i c 2bqi gt 0 for q-i lt q (a-c)/2b - q/2
and qi lt (a-c)/4b q/4 - Hence, strategies qi lt (a-c)/4b q/4 can be
eliminated at this round - Similarly for odd rounds for n 2
- Thus, for n2 game is dominance solvable
16Tilman Borgers Maarten C.W. Janssen
- This result is due to the way of replication of
the economy, i.e., only the supply side is
replicated for fixed demand - Another (more natural?) way to replace is to say
that if for a given n, demand is given by P a
bQ, then we replicate both sides and say that
for k 2,3,4, there are nk firms and demand is
P a bkQ - Their main result then says that for general cost
and demand functions and for large k, the Cournot
game is dominance solvable iff the Cobweb process
is dynamically stable
17Relation between IEDS and naïve learning
- Dominance solvability is highly sophisticated
eductive reasoning process (CKR) - Cobweb process based on very naïve learning as
also used by Cournot in creating a dynamic story
to his theory - Relation in three parts
- IEDS and rationalizability
- Rationalizability and Cournot dynamics
- Cournot for large N and Cobweb
18Play Guessing Game
- Players have to write down a number between 1 and
100 - I calculate average and take 2/3 of it
- Player that is closest to 2/3 of the average wins
- Which number do you choose?
- Analyze the game with dominated strategies
19Rationalizability
- Let S0i Si and define iteratively for all
players i Sni si ? Sn-1i ? there is a s-i ?
product of the convex hull of Sn-1-i s.t. ui(si
, s-i ) ? ui(si, s-i) for all si ? Sn-1i - The set of rationalizable strategies is R?i ?n
Sni - Intuitively, a strategy is rationalizable if it
is a best response to some belief about the other
players play, which should be a best response to
some belief the other players have about what
their opponents will do, etc. - Convex hull is used to smoothen the process and
is needed as generally, even though two pure
strategies themselves can be a best response, a
mixture of them is not - The set of rationalizable strategies is non-empty
20Rationalizability and IEDS
- If a strategy is dominated, it is never a best
response and therefore cannot be rationalizable.
Thus, rationalizability is a stronger notion - For n2, the two actually coincide
- Graphical illustration
- For N gt 2, rationalizability is stronger
- Exercise 2.7 FT