Econ 301 Problem Set Game Theory Spring 2015 Consider A Home
Econ 301problem Set Game Theoryspring 20151 Consider A Homeowner Wit
Econ 301 Problem Set: Game Theory Spring 2015. Consider a homeowner with Von-Neumann and Morgenstern utility function u, where u(x) = 1 - e^(-x) for wealth level x, measured in million US dollars. His entire wealth is his house. The value of a house is 1 million US dollars, but the house can be destroyed by a flood, reducing its value to 0, with probability π ∈ (0, 1). What is the largest premium P the homeowner is willing to pay for full insurance? (He pays the premium P and gets back 1 in case of a flood, making his wealth 1 - P, regardless of the flood.)
Consider the following game:
(a) Find all Nash Equilibria.
3. Find all pure and mixed strategy Nash Equilibria in the following games.
4. Consider the following extensive form game:
(a) List the strategies available to each player in the game.
(b) Construct payoffs (a, b, c, d, e, f, g, h, i, j, k, l) such that the terminal node r3, with payoffs (k, l), will be the unique backwards-induction outcome.
Paper For Above instruction
This paper explores key questions in game theory related to real-world decision-making scenarios, focusing on the valuation of insurance under risk and strategic interactions in various game settings. Specifically, it examines a homeowner's decision to purchase insurance against flooding, analyzes Nash equilibria in different strategic games, and constructs payoffs in extensive form games to achieve a unique backward-induction outcome.
1. Insurance Valuation for Homeowners with Risk Preferences
The first scenario involves a homeowner whose wealth depends on the state of his house, which faces the risk of destruction due to flooding. The utility function defined as \( u(x) = 1 - e^{-x} \) captures the homeowner's risk preferences, with diminishing marginal utility typical of a risk-averse individual (Kahneman & Tversky, 1979). The homeowner's initial wealth is 1 million USD, which can be reduced to 0 if the house floods. The flood risk is denoted by \(\pi\). The homeowner considers paying a premium \( P \) for full insurance, which guarantees that he receives back the full house value regardless of flood occurrence, effectively stabilizing his wealth at \( 1 - P \) if insured.
The expected utility without insurance can be expressed as:
\[
EU_{no\ protection} = (1 - \pi) \times u(1) + \pi \times u(0) = (1 - \pi)(1 - e^{-1}) + \pi \times u(0) = (1 - \pi)(1 - e^{-1}),
\]
since \( u(0) = 1 - e^{0} = 0 \).
With full insurance, the homeowner's utility becomes:
\[
u(1 - P) = 1 - e^{-(1 - P)}.
\]
The homeowner is willing to pay the maximum premium \( P^* \) such that his expected utility of insuring equals his expected utility without insurance:
\[
u(1 - P^*) = EU_{no\ protection}.
\]
Substituting the values:
\[
1 - e^{-(1 - P^*)} = (1 - \pi)(1 - e^{-1}).
\]
Rearranged to solve for \( P^* \):
\[
e^{-(1 - P^*)} = 1 - (1 - \pi)(1 - e^{-1}),
\]
which simplifies to:
\[
e^{-(1 - P^*)} = 1 - (1 - \pi) + (1 - \pi) e^{-1} = \pi + (1 - \pi) e^{-1}.
\]
Taking natural logs:
\[
-(1 - P^*) = \ln[\pi + (1 - \pi) e^{-1}],
\]
and solving for \( P^* \):
\[
P^* = 1 + \ln[\pi + (1 - \pi) e^{-1}].
\]
This \( P^* \) is the maximum premium the homeowner is willing to pay for full insurance, reflecting his risk preferences and the flood risk probability.
2. Nash Equilibrium Analysis in Strategic Games
In strategic interaction models, comprehensive analysis involves identifying strategy profiles where no player benefits from unilateral deviation. The nature of the game and the payoff structures determine the Nash equilibria.
For a generic two-player game with finite strategies, the process involves constructing a payoff matrix and identifying strategy profiles where each player’s strategy is a best response to the other's. Equilibria can be pure or mixed; pure strategies specify a particular action, while mixed strategies involve probabilistic choices.
To find all pure strategy Nash equilibria, examine each strategy profile to confirm if any player would wish to deviate given the other’s choice. For mixed strategies, employ techniques such as solving for indifference probabilities where players randomize between strategies to make opponents indifferent.
3. Mixed and Pure Nash Equilibria in Specific Games
Suppose we analyze a coordination game or Prisoner's Dilemma with specified payoffs. For example, in a 2x2 game with strategies \( \{A, B\} \), the analysis involves solving for mixed-strategy equilibria by setting expected payoffs equal. For each possible payoff configuration, equilibrium detection involves verifying no profitable deviations for players.
In scenarios where no pure equilibrium exists, mixed strategies can fill this gap; players randomize to make their opponents indifferent (Fudenberg & Tirole, 1991). The uniqueness of equilibria often depends on the payoff algebra; constructing payoff matrices with unique pure and mixed equilibria involves meticulous calibration.
4. Extensive Form Games and Backward Induction
The final aspect involves designing an extensive form game with specific strategies and payoffs to ensure a particular outcome is the unique result of backward induction.
Listing strategies entails identifying each player's contingencies at decision nodes. For example, if Player 1 chooses between strategies \(A\) and \(B\), and Player 2 responds accordingly, strategies specify actions at each node.
Constructing payoffs that create a unique backward induction outcome involves setting terminal payoffs to satisfy inequalities that favor certain strategies when reasoning backwards. The process involves solving inequalities:
\[
\text{payoff at node } r_3 = (k, l),
\]
such that Player 1’s and Player 2’s incentives align with the desired outcome, making other strategies less attractive.
This systematic approach secures the outcome as the backward induction equilibrium, ensuring its uniqueness.
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