**Exercise 1- ****Consider the following game - called "Stone-Scissor-Paper":**

There are 2 players, player 1 and 2. The two players are placed with their front against each other, with the one fist clenched. Simultanously the players rais their arm and counts to 3. Reaching 3 each player forms the fist of either: "Stone" (the fist is kept clenched), "Scissor" (the fore- and ring finger is unfolded and separated) or "Paper" (The entire fist is unfolded). The result is the: either player 1 wins, player 2 wins or it is a draw. A player who wins gets a payoff of 1, the looser gets a payoff of -1 while a draw yields a player a payoff of 0. Who wins depends on the ordering: "Stone beats Scissor", "Scissor beats Paper" and "Paper beats Stone".

1. Formulate this as a game on normalform. Find the Bi-matrix.

2. Do there exists a Nash equilibrium in pure strategies of this game?

3. What is a mixed strategi of player 1 in this game?

**Exercise 2 **

Consider the following game given by their Bimatrix

L C R

T 2,0 1,1 4,2

M 3,4 1,2 2,3

B 1,3 0,2 3,0

- Is the strategy "B" a dominated strategy for player 1?
- Find the strategies that survives an iterativ deletion of dominated strategies
- Find Nash equilibrium i pure strategies

Consider the following description of a game.

There are 2 players, player 1 and 2, and a gamemaster. The gamemaster has a coin that is bent such that, flipped randomly, the coin will come up with "heads" 80% of the time. Both players know this. The game-master flips the coin and shows the result to player 1. Player 1 makes an announcement to player 2 what the result of the flip is, either "head" or "tail". Player 2 having heard player l's announcement but not seen the result must guess what the result of the flip was. Player 1 has the following payoffs: if player 2 guess "head" payoff is 2 and payoff is 0 if the guess is "tail". Moreover, if player l's announcement is correct player l's payoff is increased by 1. For player 2 the payoff is 1 if he guesses right and 0 if his guess is wrong.

- Formulate this game as a game on extensivform. What is the game tree.
- What is the normalform representation of this game?
- Find all Nash equilibria of this game

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