SMU H3 Map

• Content map: SMU H3 Game Theory Map

Ultimatum Game

Definition

A negotiation where one player makes a “take it or leave it” offer to another player.

• The person who offers is the proposer
• The person who answers is the recipient

Game Tree

• Games can be represented as a combination of every choice made with every payoff made

• Given the game tree below, we can conduct backward induction to identify the rollback equilibrium

Rollback Equilibrium of Player 2

• Starting from the left subtree, the best outcome for this subtree is (0, 2)

• For the right subtree, the best outcome for this subtree is (1, 4)

Strategies for a Player

• P2’s strategies are a choice at each node
• Thus, the table below represents all the strategies for P2
  • (u, u)
  • (u, d)
  • (d, u)
  • (d, d)

Game

Rules

• The proposer chooses how many coins to give the recipient.
• The recipient can accept or reject.
• If the recipient rejects, both parties get nothing.

Ingredients

Players

• 2 players
• 1 proposer, 1 recipient

Information Structure

• One player moves first (proposer), then the second player (recipient).
• The recipient knows the proposer’s strategy when making the decision.

Strategy

• Proposer decides how much to offer.
• Recipient decides whether to accept or reject based on the amount.

Payoff

Let $(x, 10 - x)$ denote the proposed split.

• Proposer → $x$ if accepted, 0 otherwise
• Recipient → $10 - x$ if accepted, 0 otherwise

Backward Induction

Let $(x, 10 - x)$ denote the proposed split.

Case 1: If $x > 0$

• Proposer → $x$
• Recipient → $10 - x$, since $10 - x > 0$

Case 2: If $x = 0$

• Recipient → 0 if accept, 0 if reject
• Proposer → 10 if accept, 0 if reject

Plausible Scenarios

1. Recipient accepts all offers
   → Proposer’s best split: $(10, 0)$

2. Recipient accepts offers where payoff > 0
   → Proposer’s best split: $(9, 1)$

Rollback Equilibrium

Defined by the pair of strategies (one per player) that emerge from backward induction.

Scenarios

1. $(0,\ \text{accept any offer})$
2. $(1,\ \text{accept positive offers})$

Extension 1: Inequality-Averse Recipient

Now assume the recipient cares about the difference:

instead of just $10 - x$.

Strategy

• Accept → Payoff = $10 - 2x$
• Reject → Payoff = 0

Optimal Strategy

Best answer is YES when:

• $(4, 6)$
• $(3, 7)$
• $(2, 8)$
• $(1, 9)$
• $(0, 10)$

Best answer is NO when:

• $(6, 4)$
• $(7, 3)$
• $(8, 2)$
• $(9, 1)$
• $(10, 0)$

Since $10 - 2x < 0$ , when the split is $(5, 5)$ Both yes and no are equally optimal answers.

Rollback Equilibria

1. $(0,\ \text{accept any offer})$
2. $(1,\ \text{accept positive offers})$

Moral of the Story

• In labs, the split ratios go from 50:50 to 60:40
• Monetary gains and losses might not be the only factor influencing
• Neuroeconomics can help in understanding these determinants
• The Neural Basis of Economic Decision-Making in the Ultimatum Game
• https://doi.org/10.1126/science.1082976

Extension 2: Counter-Offers

• What if the recipient can make a counter-offer?

Rules

• Agreeing to $x$ in the second round is payoff equivalent to agreeing to $\delta x$ in the first round, where $0 < \delta < 1$
• The pie to split has size 1, where any real number $x \in [0, 1]$ can be offered
• This means that :
  • P1’s split is of the form $(x, 1-x)$
  • P2’s split is of the form $(y, 1-y)$

Backward Induction

Round 2

• P2 offers a split of $(1-y, y)$
  • If yes, P1 obtains $1-y$
  • If no, P1 obtains $0$
• Yes to any split since $y \in [0, 1]$, thus $1-y \in [0, 1]$
• P2 offers $(0, 1)$

Rollback to Round 1

• P1 proposes a split of $(x, 1-x)$
  • If yes, P1 obtains $1-x$
  • If no, the game moves to stage 2, P2 will get $\delta$

Conclusion

• P1’s best response is :
  • Yes for all $x$ such that $1-x \ge \delta$, which is equivalent to $x \le 1-\delta$
  • No otherwise
  • The best split P1 can offer is $(1 - \delta, \delta)$

Rollback Equilibrium

P1’s Strategy

• In round 1, offer the split $(1-\delta, \delta)$
• In round 2, accept any proposed split $(1-y, y)$

P2’s Strategy

• In round 1, offer the split $(0, 1)$
• In round 2, accept any proposed split $(x, 1-x)$ where $x \le 1-\delta$

Entry Game

Rules

• Monopolist faces a threat by a challenger
• The challenger may decide to enter the market
• Upon entry, the monopolist will decide to price aggressively or cooperatively
• Monopolist prefers no entry to entry, and collusion to competition
• Challenger prefers entry only under collusion

Game Tree

Rollback Equilibrium

• Challenger should enter the market, and cooperate to maximise payoff
• Monopolist can improve the outcome by cutting off the branch which results in the suboptimal rollback equilibrium
• This can be done by setting a precedent that they will not cooperate