S and J are trying to split 100 dollars. In bargaining round 1, S makes an offer at cost 0, proposing to keep S1 for himself and J either accepts (ending the game) or rejects. In round 2, J makes an offer at cost 10 of S2 for S and S either accepts or rejects. In round 3, S makes an offer of S3 at cost c, and J either accepts or rejects. If no offer is ever accepted, the 100 dollars goes to a third player, D.
a) if c=0, what is the equilibrium outcome? what if c=80? c=10?
b) what happens if c=0, but Jones is very emotional and would spit S's face and throw the 100 dollars to D if S proposes S=100? Assume here that S knows J's personality perfectly.
Assistance needed.
What is the "current macroeconomics situation"?
a) In this scenario, we can analyze each round and determine the equilibrium outcome for different values of c.
1) If c = 0:
- S offers S1 = $100 to J.
- J has the option to accept or reject the offer. If they accept, the game ends with J receiving $100 and S receiving $0. If they reject, the game proceeds to the next round.
- In round 2, J makes an offer of S2 for S. Since c = 0, S has no reason to reject the offer. Thus, S accepts any positive offer S2 > 0 from J.
- The equilibrium outcome in this case is J receiving $100 and S receiving $0 if J accepts S1, or J receiving S2 and S receiving (100 - S2) if J makes a positive offer in round 2.
2) If c = 80:
- S offers S1 = $100 to J.
- J, knowing that S values the money highly, has an incentive to make a positive offer to maximize their own share. Let's assume J offers S2 = $90 to S.
- S has the option to accept or reject. Since S values the money highly, they are likely to accept any positive offer S2.
- The equilibrium outcome in this case is J receiving $90 and S receiving $10 if J offers S2 = $90.
3) If c = 10:
- S offers S1 = $100 to J.
- J still has an incentive to make a positive offer, but the cost c = 10 introduces some uncertainty.
- Depending on J's risk aversion, they might offer S2 slightly lower than $90, ensuring a higher payoff for themselves.
- S, considering the cost c and the uncertainty in J's offer, has the option to accept or reject. If the offer is close to $90, S might accept to secure a positive payoff.
- The equilibrium outcome in this case depends on the specific values and negotiation strategies of S and J.
b) If c = 0 and Jones is emotional and would spit in S's face and throw the $100 to D if S proposes S = $100, the potential outcomes change. In this case, it is important for S to consider J's emotional reaction.
- S offers S1 = $100 to J, but if S proposes S = $100, J, being emotional, would reject the proposal and throw the money to D.
- Knowing J's personality, S should reason that J would be inclined to reject any offer that is close to S = $100.
- Thus, S should make an offer that is lower than $100 to avoid triggering J's emotional reaction.
- The equilibrium outcome would depend on S's negotiation strategy, but it is likely that S would propose an offer that is significantly lower than $100 to ensure J's acceptance and avoid J's emotional response.