Question

Players 1 and 2 are leaving the house and need to decide whether or not to bring an umbrella. They know there's a 75% chance that it will rain (thanks AccuWeather). If they don't bring an umbrella and it rains they receive a payoff of -7. If they do bring an umbrella and it rains, they receive a payoff of -4 . If it doesn't rain, but they bring an umbrella they receive a payoff of -2. Finally, if it doesn't rain and they don't bring an umbrella they receive a payoff of 2 . Player 1 learns the weather before they leave the house, but player 2 does not. Player 2 can, however, observe player 1 s choice.

(a) Represent this game in the extensive form.
i. Use backwards induction to solve for the Nash equilibrium

Answer 1

Using backwards induction and expected payoffs, the Nash equilibrium for the given game is that both players would opt to bring an umbrella. This decision accounts for the payoffs of each outcome and Player 1's informational advantage.

Step-by-step explanation:

The extensive form game described involves strategic decision-making under uncertainty, with player 1 having an informational advantage. Using backwards induction, we can solve for the Nash equilibrium of the game as follows:

• Player 1 will bring an umbrella if they know it will rain since the payoff is higher (-4) compared to not bringing an umbrella (-7).
• Player 2 will observe player 1's action. If Player 1 does not bring an umbrella, Player 2 deduces it won't rain and also decides not to bring an umbrella. If Player 1 brings an umbrella, Player 2 cannot deduce the weather but knows Player 1 had perfect information.
• Therefore, Player 2 may opt to follow Player 1's decision based on their informational lead.
• Using expected payoffs, if Player 1 does not know the weather, they will compare the expected payoff of bringing an umbrella (0.75*(-4) + 0.25*(-2)) to that of not bringing one (0.75*(-7) + 0.25*2).
• The expected payoff for bringing an umbrella is -3.5, whereas for not bringing it is -4.75, thus Player 1 should bring an umbrella regardless.

Consequently, the Nash equilibrium in this case would involve both players choosing to bring an umbrella regardless of the weather forecast Player 1 has received. This analysis incorporates both notions of rationality and strategy in game theory.