17. backward induction –
ultimatums and bargaining
take it or leave it offers.
Two players. Split a dollar. (s,1-s) offer to 2.
if accepts get (s,1-s). Otherwise zero for both.
Does backward induction permit (.99, .01),
(100,0)
IN splitting the dollar
• Why do we see a failure of backwards
induction? (see this happen in practice)
– pride thing
– want to be fair
Two period game. “Shrinking pie” time
of making the decision enters in
Dollar on the table.
Make an offer
Player accept/reject. If accept (s, 1-s)
If reject, flip roles – but part of the money is lost.
delta (90%, say) remains.
If player 2 offers 2 > , player 2 will accept.
Otherwise, will reject.
For k stages, we get powers which determine what
we need to offer to the other person:  k-1
Results of shrinking pie?
We get an even split if we can potentially bargain forever.
If the value of the pie and the discount is known, the first
offer will be accepted. (No haggling in equilibrium)
The poor will do less well in bargaining.
If people's values are not known, not only are the offers not
accepted immediately and not only is there some inequity
in that the poor tend to be more impatient and do less
well, but also you get bad inefficiency. The inefficiency
occurs essentially because the sellers want to seem like
they're hard bargainers and the buyers want to seem like
they're hard, and you get a failure for deals to be made.
If values are not known, offers are not accepted
on the first round.
Get bad efficiency. Both want to seem like they
are hard – failure for deals to be made.
Lecture 18 - Imperfect Information: Information
Sets and Sub-Game Perfection
• Assume perfect recall.
• Can’t assume two nodes are in same information
set, if those nodes have different number of
choices.
• One reason for an information set could be that
player 1 is randomizing between their choices.
• perfect information – all information sets have just
one node.
• Can’t fail two distinguish between nodes that
depend on YOUR prior moves. May not be true if it
is a “team” that is making a decision (and others
don’t know what another said).
What matters is information not time
Subgame
1. starts from a single node
2. comprises all successors to that node
3. does not break up any information sets
Are these subgames?