L8 - Game Theory: Bargaining II Flashcards
How is the non-cooperative approach to bargaining different to the cooperative approach?
The cooperative approach makes use of an axiomatic analysis by asking what properties we would expect a bargaining solution to exhibit. However, it makes the uncomfortable assumption that players can make binding agreements.
The non-cooperative approach instead requires any prediction to be immune from individual profitable deviations and tries to better analyse individual behaviour and interaction. However, to proceed it needs to place more structure on the specific bargaining context.
What are some of the common features of the non-cooperative approach?
To apply our equilibrium concepts, we need to place more structure on the bargaining game. This can be done in a variety of ways
. Some common themes include:
- • Players 1 and 2 have to divide a monetary amount or surplus, X>0, between themselves. (divide a pie between themselves)
- • The premise that as the longer the bargaining takes, the more time passes, the more the payoffs for both parties shrink.
- o This arises either because the surplus actually reduces (e.g. opportunities pass by) and/or because the players prefer an agreement earlier due to standard time discounting.
One particular structure seems natural and appealing:
- • The players make offers and counteroffers sequentially until an agreement is reached. Many forms of negotiation seem to fit this pattern. To analyse proceed, we’ll build up slowly step by step, first considering a very simple form called the Ultimatum game
How do we set up the Ultimatum Game?
What is the equilibria of the ultimatum game?
More precisely, if m is in increments (e.g. pounds and pence), there are actually two SPNE depending on the tie breaking rule for when P2 is indifferent given an offer m=0.00.
- 1) If P2 will reject an offer m = 0.00, then P1 offers m = 0.01 in the SPNE.
- 2) If P2 will accept and offer m = 0.00 then P1 offers m = 0.00 in the SPNE.
- In either equilibrium, P1 gains (X-m) and so keeps the majority of the surplus.
- This implies that the position of the first-proposer,
- P1, offers them a big source of bargaining power.
- Thus, unlike the cooperative approach, a source of bargaining power arises endogenously from the structure of the game
If m is measured with no increments (any real number), then there is only one formal, well-defined SPNE (as P1 will keep offering incrementally smaller values of m e.g. 0.01 no 0.001 no 0.0001 etc):
P1 offers m=0 and P2 accepts m=0
- Again, the first-proposer, P1, has a big source of bargaining power. –> major advantage
- If, instead, P2 only accepted any m>0 then P1 would always want to offer something closer and closer to zero and so P1’s strategy would not be well defined.
- Always an incentive to deviate closer to zero. We will return to the Ultimatum game later in the module when we study experimental and behavioural economics.
What is the structure of a two-period alternating offer in a non-cooperative approach?
- P2 can give a counteroffer in the second period
- discount factor describes how ‘ relative patience’ of the two players.
How do you solve the two-period alternating offer?
Period 2: (Basically, an Ultimatum game)
P1 should accept any offer m1 ≥ 0, and so P2 can offer m1 = 0 to gain everything for themselves with a payoff equivalent to 1.
Period 1:
- P1 makes an offer, m2. If P2 rejects it, P2 knows that they can get a payoff of 1 in period 2, but with a discount, this is only worth 𝜹𝟐 to them now.
- Thus, P2 will accept any offer from P1 in period 1 of m2 ≥ 𝜹𝟐.
- Hence, in period 1, P1 optimally offers m2 = 𝜹𝟐 and P2 accepts. P2 earns m2 = 𝜹𝟐 and P1 earns the remainder, 1 - m2 = 1 - 𝜹𝟐.
- Patience matters, 𝜹𝟐. If P2 is more willing to wait, her disagreement payoff is higher in a sense, and so P1 has to offer them more to accept.
- One can use many other game structures to calculate bargaining outcomes in other finite-period scenarios
- We’ll now move on to thinking about a very rich, and realistic structure where offers and counter-offers can continue indefinitely until an agreement is reached.
*
How do you set up an infinite period alternating offer game?
How do you derive the equilibrium of the Infinite period Alternative offer game?
- Payoff is based on the relative discount factors of the two players
- a player payoff is based on their own discount factor –> the more patient they are the larger their payout will be
What are the implications to the equilibrium of the infinite period alternative offer game with symmetric disagreement factors?
- P1 still gains a higher payoff than P” due to their position advantage from going first. Remember δ likes between 0 and 1.
- If δ = 0, P1 would earn everything ( offer nothing to P2 who perfectly impatient like P1 who will accept the offer straight away)
- tomorrow is worthless for both players
- collapses down to the ultimatum game again
- When 𝛿 → 1, the players are almost completely patient, they treat a pound today almost the same as a pound tomorrow - perfectly patient
This overlap can happen a little more generally too, detail not needed!
Evaluation of the Non-cooperative approach to game theory?
- Relative to the cooperative approach, the non-cooperative approach tries to model behaviour and interaction in more detail
- . It produces some useful additional insights, and it is encouraging to know that the cooperative and non-cooperative approaches can coincide. Complementary.
- However:
- • The non-coop approach produces that predictions are very sensitive to the specific context, and the specific structure of the game.
- What style of game is it, who goes first, what do they offer, what they do/dont know
- • Across structures, it also often predicts that bargaining should come to an immediate agreement in period 1. This is in stark contrast to reality where we see disagreements, strikes and stalling
- • The non-coop approach produces that predictions are very sensitive to the specific context, and the specific structure of the game.
- To understand such phenomena, one has to add asymmetric information
