Problem 1
Discuss some examples of jobs about which you know, highlighting the relation between risk, risk aversion, and the use of bonuses.
Problem 3
A firm and worker interact as follows. First, the firm offers the worker a wage \(w\) and a job \(z ; z=0\) denotes the "safe" job and \(z=1\) denotes the "risky" job. After observing the firm's contract offer \((w, z)\), the worker accepts or rejects it. These are the only decisions made by the firm and worker in this game. If the worker rejects the contract, then he gets a payoff of 100 , which corresponds to his outside opportunities. If he accepts the job, then the worker cares about two things: his wage and his status. The worker's status depends on how he is rated by his peers, which is influenced by characteristics of his job as well as by random events. Specifically, his rating is given by \(x\), which is either 1 (poor), 2 (good), or 3 (excellent). If the worker has the safe job, then \(x=2\) for sure. In contrast, if the worker has the risky job, then \(x=3\) with probability \(q\) and \(x=1\) with probability \(1-q\). That is, with probability \(q\), the worker's peers think of him as excellent. When employed by this firm, the worker's payoff is \(w+v(x)\), where \(v(x)\) is the value of status \(x\). Assume that \(v(1)=0\) and \(v(3)=100\), and let \(y=v(2)\). The worker maximizes his expected payoff. The firm obtains a return of \(180-w\) when the worker is employed in the safe job. The firm gets a return of \(200-w\) when the worker has the risky job. If the worker rejects the firm's offer, then the firm obtains 0 . Compute the subgame perfect equilibrium of this game by answering the following questions.(a) How large must the wage offer be in order for the worker rationally to accept the safe job? What is the firm's maximal payoff in this case? The parameter \(y\) should be featured in your answer. (b) How large must the wage offer be in order for the worker rationally to accept the risky job? What is the firm's maximal payoff in this case? The parameter \(q\) should be featured in your answer. (c) What is the firm's optimal contract offer for the case in which \(q=1 / 2\) ? Your answer should include an inequality describing conditions under which \(z=1\) is optimal.
Problem 5
5\. Consider a game inspired by the long-running television program The Price Is Right. Three players have to guess the price \(x\) of an object. It is common knowledge that \(x\) is a random variable that is distributed uniformly over the integers \(1,2, \ldots, 9\). That is, with probability \(1 / 9\) the value of \(x\) is 1 , with probability \(1 / 9\) the value of \(x\) is 2 , and so on. Thus, each player guesses a number from the set \(\\{1,2,3,4,5,6,7,8,9\\}\). The game runs as follows: First, player 1 chooses a number \(n_{1} \in\\{1,2\), \(3,4,5,6,7,8,9\\}\), which the other players observe. Then player 2 chooses a number \(n_{2}\) that is observed by the other players. Player 2 is not allowed to select the number already chosen by player 1 ; that is, \(n_{2}\) must be different from \(n_{1}\). Player 3 then selects a number \(n_{3}\), where \(n_{3}\) must be different from \(n_{1}\) and \(n_{2}\). Finally, the number \(x\) is drawn and the player whose guess is closest to \(x\) without going over wins \(\$ 1000\); the other players get 0 . That is, the winner must have guessed \(x\) correctly or have guessed a lower number that is closest to \(x\). If all of the players' guesses are above \(x\), then everyone gets \(0 .\) (a) Suppose you are player 2 and you know that player 3 is sequentially rational. If player 1 selected \(n_{1}=1\), what number should you pick to maximize your expected payoff? (b) Again suppose you are player 2 and you know that player 3 is sequentially rational. If player 1 selected \(n_{1}=5\), what number should you pick to maximize your expected payoff? (c) Suppose you are player 1 and you know that the other two players are sequentially rational. What number should you pick to maximize your expected payoff?
Problem 6
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 \(\mathrm{A}\) and with probability \(2 / 3\) the coin stays in cell \(\mathrm{C}\). The coin cannot go beyond an end zone. Thus, if player 1 selects Slap when the coin is in cell \(\mathrm{D}\), then the coin moves to cell \(\mathrm{E}\) with probability \(1 / 3\) and it remains in cell \(\mathrm{D}\) 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 \(v^{k}\) 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 \(w^{k}\) 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.
