Consider a game in which two players take turns moving a coin along a strip of wood (with borders to keep the coin from falling off). The strip of wood is divided into five equally sized regions, called (in order) A, B, C, D, and E. Player l's objective is to get the coin into region E (player 2's end zone), whereas player 2's objective is to get the coin into region A (player l's end zone). If the coin enters region A, then the game ends and player 2 is the winner. If the coin enters region E, then the game ends and player 1 is the winner. At each stage in the game, the player with the move must decide between "Push" and "Slap."
If she chooses Push, then the coin moves one cell down the wood strip in the direction of the other player's end zone. For example, if the coin is in cell B and player 1 selects Push, then the coin is moved to cell C. If a player selects Slap, then the movement of the coin is random. With probability 1/3, the coin advances two cells in the direction of the other player's end zone, and with probability 2/3 the coin remains where it is. For example, suppose it is player 2's turn and the coin is in cell C. If player 2 chooses Slap, then with probability 1/3 the coin moves to cell A and with probability 2/3 the coin stays in cell C.
The coin cannot go beyond an end zone. Thus, if player 1 selects Slap when the coin is in cell D, then the coin moves to cell E with probability 1/3 and it remains in cell D with probability 2/3. If a player wins, then he gets a payoff of 1 and the other player obtains zero. If the players go on endlessly with neither player winning, then they both get a payoff of 1/2. There is no discounting. The game begins with the coin in region B and player 1 moves first.
(a) Are there any contingencies in the game from which the player with the move obviously should select Push?
(b) In this game, an equilibrium is called Markov if the players' strategies depend only on the current position of the coin rather than on any other aspects of the history of play. In other words, each player's strategy specifies how to behave (whether to pick Push or Slap) when the coin is in cell B, how to behave when the coin is in cell C, and so on.
For the player who has the move, let vk denote the equilibrium continuation payoff from a point at which the coin is k cells away from this player's goal. Likewise, for the player who does not have the move, let wk denote the equilibrium continuation payoff from a point at which the coin is k cells away from this player's goal. Find a Markov equilibrium in this game and report the strategies and continuation values. You can appeal to the one-deviation property.