Download Bargaining games

Bargaining games
Econ 414
General bargaining games
• A common application of repeated games is to
examine situations of two or more parties
bargaining over a payment (or division of some
total surplus).
• In each “round”, a player makes an offer, which
the other party may accept (in which case the
game stops) or reject (in which case the game
continues). The game continues for some given
number of rounds (which could be infinite), but
the surplus decreases in value each round,
because of bargaining costs or depreciation.
Example: Alternating offer
bargaining
• The most common bargaining game has two players who alternate
offers each round. There is a total surplus S which players are
bargaining over, and the value of S drops by a multiplicative
discount factor δ each period.
• The game works as follows: Player 1 proposes some offer x1 to
player 2. Player 2 may then accept (in which case payoffs are S x1, x1) or reject the offer (in which case the game continues).
Player 2 then proposes an offer y1 to player 1. Player 1 may accept
the offer (payoffs are then δy1, δ(S – y1)) or reject the offer, in which
case the game continues.
Player 1 then gets to offer again, but over δ2S, which if rejected
player 2 may make an offer over δ3S.
• This alternating offer process then continues until an offer is
accepted.
1
x1
2
R
A
S – x1, x1
2
y1
1
A
δy1, δ(S - y1)
R
1
x1
2
Finite length
• If the game has a finite number of periods T (ie T total offers), then
we have a terminal payoff if the offer in the final round is rejected,
which is typically 0,0.
• Then the game can be solved as a regular dynamic game, finding a
subgame perfect nash equilibrium by backward induction.
• The trick is to note that in each period, the player making the offer
will offer just enough such that the player receiving the offer is
indifferent between accepting and rejecting the offer (and we will
assume that they will accept it).
• Thus, in the final period, the player making the offer will make an
offer of zero, keeping δT-1S for themselves.
In the preceding period, the player making the offer will offer just
enough to make the receiving player accept – ie make an offer so
that their payoff is δT-1S.
• This pattern then continues back to the first period. So the result of
the game is that the initial offer at the beginning of the game will be
accepted, but the offer will depend on how many rounds we have.
Example, T = 4
•
•
•
•
•
•
In round 4, 1 gets 0 if they reject the offer, so they accept any non-negative
offer, so 2 offers them y2 = 0.
This gives 2 a round 4 payoff of δ3S.
In round 3, 1 must make an offer that leaves 2 indifferent between accepting
that offer and rejecting the offer (which gives δ3S). So 1 will offer x2 = δS.
In round 2, 2 must make an offer that leaves 1 indifferent between accepting
that offer and rejecting the offer (which gives δ2(S- δS)). So 2 will offer y1 =
δS(1- δ) = δS- δ2S.
In round 1, 1 must make an offer that leaves 2 indifferent between accepting
that offer and rejecting the offer (which gives δ[S – δS(1- δ)]). So 1 will offer
x1 = δ[S – δS(1- δ)] = δS – δ2S + δ3S.
The unique SPNE then is for players to make the offers xi, yi above in the
appropriate periods, and to accept any offer of at least xi, yi above in the
appropriate periods.
Thus, in the unique equilibrium, no bargaining actually occurs; the initial
offer is accepted immediately.
The pattern in this game continues. If we had T = 6, then x1 would be δS –
δ2S + δ3S - δ4S + δ5S
T=∞
• Now, suppose the game is repeated an infinite
number of times. We cannot use a normal
backward induction process, because there is
no last period to solve from.
• However, there is a recurring pattern in the
payoffs and strategies (the game is “stationary”)
that we can exploit to find the equilibrium offers.
• It turns out that the unique equilibrium is one
where each player makes the same offer every
period (which is the highest that would be
accepted in that period). I will not prove this
here.
• Assume the form of the solution: that player 1 offers x*
each period and player 2 offers y* each period.
Start at any period N where player 1 makes an offer of
x*. Then, if player 2 accepts, they get δN-1x*. If they
reject, 2 will then offer y*, which will be accepted, so 2
will get δN(S – y*). Thus, these must be equal, so x* =
δ(S – y*).
• Similarly, stating at a period N+1 where player 2 makes
an offer of y*, player 1 gets δNy*. If player 1 rejects, then
1 will offer x* which will be accepted, giving 1 a payoff of
δN+1(S – x*). These must be equal, so y* = δ(S – x*).
• Solve these two bold equations simultaneously, gives x*
= y* = δS/(1 + δ).
Convergence
• The solution to the T-length finite game
(for the initial bid) converges to the
solution for the infinitely repeated game as
T -> ∞.
• δS – δ2S + δ3S - δ4S + δ5S + ….
= δS/(1 – δ2) – δ2S/(1 – δ2)
= δS(1 – δ)/(1 – δ2)
= δS/(1 + δ)
Alternate notation
• There are other common forms of notation for bargaining
models which are conceptually identical, but behave
slightly differently algebraically.
• Osborne for eg has players make an offer a pair (x1, x2)
where x1 is the payoff to player 1 and x2 is the payoff to
player 2. Implicitly, x2 = S – x1 in my notation, and S is
normally assumed to be 1.
• Another common notation is again to implicitly assume
that S = 1, but to make all offers the payoff to player 1.
So offer x1 if accepted gives x1, 1 – x1. y1 if accepted
gives δy1, δ(1 – y1). X2 if accepted gives δ2x2, δ2(1 – x2).
This is functionally the same, but gives us different
values for x* and y* (y* = 1 – x*).
Reality?
• The common feature of nearly all bargaining games is
that despite a sometimes complex strategy structure,
the game is resolved at the very beginning, when the
initial offer is made and accepted. Thus, no bargaining
process is ever actually observed in play of the game.
• But in reality, we often observe protracted bargaining
behavior (union negotiations and strikes, competing
takeover offers, almost any actual bargaining scenario).
• So, what is going on? Are people irrational (in the strict
game theory sense)? Maybe we have misrepresented
preferences or information in the game?