John Carey, a student studying at Johns Hopkins business school, has a credit card account with VISA. VISA’s available strategies are to raise Carey’s credit card interest rate or do nothing. Carey’s available strategies are to transfer his VISA account balance to another creditor or do nothing. The payoffs are the following:

If Carey does nothing and VISA does nothing, each player receives $0.

If Carey does nothing and VISA raises Carey’s interest rate, VISA earns profits of $1,000 while Carey receives -$1,000.

If Carey transfers his account to another creditor and VISA does nothing, VISA receives -$300 while Carey receives -$150.

If Carey transfers his account to another creditor and VISA raises Carey’s interest rate, VISA receives -$350 while Carey receives -$100.

Assuming that Carey moves first, what is the Nash equilibrium in this sequential-move game? Hint: I would draw the game tree to answer this question

Question

John Carey, a student studying at Johns Hopkins business school, has a credit card account with VISA. VISA’s available strategies are to raise Carey’s credit card interest rate or do nothing. Carey’s available strategies are to transfer his VISA account balance to another creditor or do nothing. The payoffs are the following:

If Carey does nothing and VISA does nothing, each player receives $0.

If Carey does nothing and VISA raises Carey’s interest rate, VISA earns profits of $1,000 while Carey receives -$1,000.

If Carey transfers his account to another creditor and VISA does nothing, VISA receives -$300 while Carey receives -$150.

If Carey transfers his account to another creditor and VISA raises Carey’s interest rate, VISA receives -$350 while Carey receives -$100.

Assuming that Carey moves first, what is the Nash equilibrium in this sequential-move game? Hint: I would draw the game tree to answer this question

BlackTom AI · Accepted Answer

Because this is a sequential game with Carey moving first, use backward induction to find the subgame-perfect Nash equilibrium.

Relevant concepts: sequential (extensive-form) games, backward induction, best responses, subgame-perfect Nash equilibrium (strategies must be optimal at every decision node).

Step-by-step reasoning and calculations:
1. List payoffs for each terminal history:
   - Carey: Do nothing; VISA: Do nothing → Carey 0, VISA 0.
   - Carey: Do nothing; VISA: Raise → Carey -1000, VISA 1000.
   - Carey: Transfer; VISA: Do nothing → Carey -150, VISA -300.
   - Carey: Transfer; VISA: Raise → Carey -100, VISA -350.

2. Determine VISA’s best response at each of its decision nodes (i.e., after each possible Carey move):
   - After Carey chooses Do nothing: VISA compares Do nothing (payoff 0) vs Raise (payoff 1000). 1000 > 0, so VISA’s best response is Raise.
   - After Carey chooses Transfer: VISA compares Do nothing (payoff -300) vs Raise (payoff -350). -300 > -350, so VISA’s best response is Do nothing.

3. Carey anticipates VISA’s responses and chooses the initial move that gives him the higher payoff:
   - If Carey chooses Do nothing, VISA will Raise → Carey gets -1000.
   - If Carey chooses Transfer, VISA will Do nothing → Carey gets -150.
   - Compare -1000 vs -150: -150 is better for Carey, so he chooses Transfer.

Why the correct answer is correct:
- The backward-induction outcome is (Carey: Transfer; VISA’s action after Transfer: Do nothing). On the equilibrium path VISA does nothing while Carey transfers his balance. Both players’ actions are best responses at their nodes, so this is subgame-perfect.

Why other options are incorrect:
- (Do nothing, Do nothing) fails because if Carey does nothing, VISA prefers Raise (1000>0).
- (Raise interest rate, Transfer) would have VISA raising after Transfer, but VISA prefers Do nothing then (-300>-350).
- (Raise interest rate, Do nothing) is inconsistent: if Carey does nothing VISA prefers Raise; Carey would not choose Do nothing knowing VISA will Raise (-1000 vs -150).
- “There is no equilibrium” is false because backward induction yields a clear equilibrium.

Conclusion: The Nash equilibrium is (VISA, Carey) = (Do nothing, Transfer), matching the provided answer.

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