Analyze a negotiation scenario where 3 rational players attempt to divide a shared resource through strategic bidding and renegotiation | Step-by-Step Solution

What We're Solving:

We're analyzing a three-player bargaining game where players must strategically negotiate to divide 100 cents, knowing that any two players can form a coalition, but the third player can always counter-propose. This is a classic problem in cooperative game theory!

The Approach:

The key insight here is understanding backward induction and coalition stability. We need to think about what each player's best response would be at every stage, working backwards from potential final outcomes. The fascinating part is that this creates a dynamic where players must anticipate not just immediate responses, but chains of counter-proposals!

Step-by-Step Solution:

Step 1: Identify the Core Concept This is a variation of the "divide the dollar" game with majority rule. Since any 2 out of 3 players can agree on a split, we have a majority voting game with transferable utility.

Step 2: Find the Minimum Winning Coalition

• Any 2 players form a winning coalition
• The third player has no power once 2 others agree
• The excluded player can try to "buy off" one member of the current coalition

Step 3: Apply Backward Induction Logic Let's say Players A and B initially agree to split 50-50, leaving Player C with nothing.

• Player C can offer Player A (or B) 51 cents to defect from the AB coalition
• Player A would accept since 51 > 50
• But now Player B can offer Player A 52 cents... and so on!

Step 4: Recognize the Theoretical Problem This creates an infinite regression in pure theory - each player can always offer slightly more to break up any coalition.

Step 5: Find the Stable Solution The Shapley Value provides the stable solution:

• Each player has equal probability of being the "swing vote"
• Each player's expected payoff = 100/3 ≈ 33.33 cents
• This represents the fair division based on each player's marginal contribution

The Answer:

The stable outcome is that each player receives approximately 33.33 cents. This occurs because in a symmetric three-player majority game, each player has equal bargaining power despite the majority rule structure.

Memory Tip:

Think of it as a "musical chairs" situation - while only 2 players can sit at any moment, all 3 players are equally likely to end up standing! The threat of being excluded gives each player exactly 1/3 bargaining power. 🎵

Great job tackling this complex game theory problem! The beauty of this scenario is how it shows that potential coalitions can be just as powerful as actual coalitions in determining outcomes.