George and Gina:  A Match Made in Heaven? Subgame Perfection

George and Gina are sweethearts.  After dating for a mere ten years, George must decide on a location for his anniversary dinner with Gina.  Once the choice is made it is known to both of them.  The choices are Perrier, a very up-scale restaurant in Philadelphia, and the local McDonald's.

Being a very deliberate fellow, George must also decide beforehand on appropriate dinner conversation for each venue. If he takes Gina to Perrier then George feels that he must either formally propose, or prevaricate and postpone the big announcement for another year. If he takes Gina to McDonald's then George can still propose, but if he chooses not to propose then he better have something of interest to Gina that he can talk about, namely stocks.  George's topic of conversation will not be known to Gina until they are in the restaurant.

Gina thinks it is about time that she and George settle on a future for their relationship.  Indeed, she has decided that she will preempt George's chosen topic and either accept or refuse his offer of marriage should she find herself dining at Perrier.  It is so inconceivable to her that George would be so crass as to propose in McDonald's that she will prepare herself to discuss either football or bonds should she find herself there.

The payoffs to George and Gina are shown in the following game tree.

What are the subgames in this marriage game between George and Gina? There is one at the node labeled P&P.  Fill in the payoffs and find the Nash equilibria.

 Subgame at Propose and Prevaricate (P & P) Gina Accept Refuse George Propose Prevaricate

There is another subgame at the node labeled P & S.  Fill in the payoffs and find the Nash equilibria.

 Subgame at Propose and Stocks (P & S) Gina Football Bonds George Propose Prevaricate

The entire game is also a subgame.  Fill in the remaining payoffs and find the Nash equilibria. In working with the following normal form game recall that a strategy is a complete contingent plan.  This means that George must have a plan of action in the event that he finds himself at any node in the game.  Therefore each of his strategies consists of three possible moves.

 The Normal Form of George's and Gina's Mating Dance Gina Strategy 1 Strategy 2 Strategy 3 Strategy 4 At P&P At P&S At P&P At P&S At P&P At P&S At P&P At P&S Accept Foot Refuse Foot Accept Bonds Refuse Bonds George Perrier Propose Propose 3, 8 1, 2 Perrier Prevar Propose 7, 9 Perrier Propose Stocks 3, 8 Perrier Prevar Stocks 7, 9 McD Propose Propose 10, 4 10, 4 8, 3 8, 3 McD Prevar Propose 10, 4 McD Propose Stocks 0, 5 McD Prevar Stocks 0, 5 4, 0 At Rest At P&P At P&S

You should have found a Nash equilibrium in each of the two subgames.  In the normal form of the entire game there is only one subgame perfect equilibrium.  Explain why one of the subgame Nash equilibria is not a subgame perfect equilibrium.