How to static/dynamic complete info

Question 1

Mixed equilibrium (BR)

Answer 1

In general, we want to find the probability, which make the other player indifferent between his actions.

1. P1 weakly prefers y given the chance q of P2 playing their action.
2. Calculate P1´s BR to q procent of P2 choosing z. The calculated q make P1 indifferent between playing y or x.
3. Do the same for P2 for BR to p procent.
4. Show both BR, where you have to be obs on how they write the restrictions.
5. Draw a figure.

Question 2

BR-game, NE

Answer 2

When reading the question, notice if there is symmetry between the players payoff functions. If there is symmetry we only have to solve for one of the players BR, because they would be alike.

1) FOC wrt. x_i, where we isolate x_i and get the BR(x_i)=R_i(x_i) for one or both of the players.

2) Draw the graph, by plotting in 0 up to 1 in x_i place, then you would see the constraints

Question 3

BR-game, SPNE

Answer 3

Solve by backwards induction

1) Plug in P2 BR in P1´s payoff function and isolate the optimal x

2) Plug the optimal x into the what becomes the optimal BR function.

3) Thus we have one or more SPNE where the optimal x is …