Compute Markov steady state and expectations

Q: Compute Markov steady state and expectations

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Question

Probability and Game Theory: Three Sub-questions

Context: The exact transition matrix P (for Q1) and the 2×2 payoff matrix (for Q3) are not provided. Below, you will (a) solve generically in symbolic form, and (b) see a small numeric example to illustrate the procedure.

Finite-state Markov chain

Given a finite-state, irreducible, aperiodic Markov chain with transition matrix P, compute its stationary distribution π by solving πP = π with ∑ᵢ πᵢ = 1. Explain why the solution is unique.

Expectations from first principles

Let X ~ Exponential(λ) and N ~ Poisson(λ) with λ > 0. Derive E[X] and E[N] from first principles (integration/summation), showing intermediate steps.

2×2 zero-sum game

For a 2×2 zero-sum game with Player A’s payoff matrix [ [a, b], [c, d] ], find the mixed-strategy Nash equilibrium. Report the probability each player assigns to their first action (row 1 for A, column 1 for B), and justify the equilibrium with probability calculations.

Compute Markov steady state and expectations

Probability and Game Theory: Three Sub-questions

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