Sequential Bargaining and Voting Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Sequential Bargaining and Voting Game

• Players: Five pirates ordered by seniority: $A$, $B$, $C$, $D$, and $E$.
• Strategies: A proposer chooses an allocation of 100 coins among the survivors; Each voter chooses Yes or No after seeing the proposal; Preferences are lexicographic: Survival, Coins, Fewer Rivals.

Rules

• Start with five pirates dividing $100$ coins by seniority; Players alternate proposing allocations and voting Yes or No among survivors.
• The player who forms a passing coalition survives with the proposed allocation.
• If a proposal passes, the game ends; If it fails, the proposal is killed and the next pirate proposes.
• If a tie in payoffs occurs, the proposer is accepted
• With $n$ survivors, the proposer needs only enough votes to avoid losing.

Game Tree

Legend:

• Pass (P): Proposal accepted
• Fail (F): Proposer killed

Payoff Details

\renewcommand{\arraystretch}{1.3} Survivors & Rollback proposal \ $E$ & $(0,0,0,0,100)$ \ $D,E$ & $(0,0,0,100,0)$ \ $C,D,E$ & $(0,0,99,0,1)$ \ $B,C,D,E$ & $(0,99,0,1,0)$ \ $A,B,C,D,E$ & $(98,0,1,0,1)$

• Each vector is written in the order $(A,B,C,D,E)$.

Derivation (Backward Induction)

Step 1: Only $E$ survives

• No vote is needed; $E$ keeps all 100 coins, so the rollback outcome is $(0,0,0,0,100)$.

Step 2: $D$ proposes to $E$

• If $D$ is killed, Step 1 gives $E=100$.
• $D$ votes Yes on his own proposal; a $1$-$1$ tie passes, so $D$ needs no extra vote.
• $D$ maximises own coins by proposing $(0,0,0,100,0)$; $E$ can vote No, but the proposal still passes.

Step 3: $C$ proposes to $D,E$

• If $C$ is killed, Step 2 gives continuation payoffs $(0,0,0,100,0)$.
• $D$ gets 100 after rejection, so rejects any offer below 100; $E$ gets 0, so 1 coin makes $E$ prefer acceptance.
• With three survivors, $C$ needs one extra vote besides his own; the cheapest pivotal vote is $E$ for 1 coin.
• $C$ proposes $(0,0,99,0,1)$.

Step 4: $B$ proposes to $C,D,E$

• If $B$ is killed, Step 3 gives continuation payoffs $(0,0,99,0,1)$.
• $C$ gets 99 after rejection, so needs more than 99; $E$ gets 1, so needs more than 1.
• $D$ gets 0 after rejection, but 0 now ties on coins and rejection leaves fewer rivals, so $D$ needs 1.
• With four survivors, $B$ needs one extra vote because a $2$-$2$ tie passes; the cheapest pivotal vote is $D$ for 1 coin.
• $B$ proposes $(0,99,0,1,0)$.

Step 5: $A$ proposes to $B,C,D,E$

• If $A$ is killed, Step 4 gives continuation payoffs $(0,99,0,1,0)$.
• $B$ gets 99 after rejection, so needs more than 99; $D$ gets 1, so needs more than 1.
• $C$ and $E$ each get 0 after rejection, so each can be bought with 1 coin.
• With five survivors, $A$ needs two extra votes besides his own; the cheapest pivotal votes are $C$ and $E$ for 1 coin each.
• $A$ proposes $(98,0,1,0,1)$.

Derivation (Nash Equilibrium)

• With all five pirates, $A$ needs two extra votes.
• The cheapest votes are $C$ and $E$, so $A$ proposes

• $A$, $C$, and $E$ vote Yes, so the proposal passes $3$-$2$.

Nash Equilibrium

Result:

The rollback equilibrium proposal is

It passes immediately with votes from $A$, $C$, and $E$.

Social Optimum

• Total coins always sum to 100.
• The game is about survival and distribution, not surplus creation.

Insights

Insight:

• Backward induction turns the five-pirate problem into a chain of smaller voting games.
• A proposer buys the cheapest pivotal votes, not the friendliest pirates.
• The tie rule is decisive because it lowers the number of votes the proposer must buy.