**赛局理论**（英语：Game Theory），又译为**对策论**或**赛局理论**，是经济学的一个分支，1944年冯·诺伊曼与奥斯卡·摩根斯特恩合著《博弈论与经济行为》，标志著现代系统博弈理论的的初步形成，因此他们被称为“博弈论之父”。博弈论被认为是20世纪经济学最伟大的成果之一。目前可以应用在生物学、经济学、国际关系、计算机科学、政治学、军事战略，研究**游戏**或者**博弈**内的相互作用，是研究具有斗争或竞争性质现象的数学理论和方法。也是运筹学的一个重要学科。 现代的赛局理论的源头是约翰·冯·诺伊曼对于双人零和赛局的混合策略均衡点的发想和证明

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Exercise 1.1. Antonia and Bob cannot decide where to go to dinner. Antonia proposes the following procedure: Antonia will write on a piece of paper either the number 2 or the number 4 or the number 6, while Bob will write on his piece of paper either the number 1 or 3 or 5 . They will write their numbers secretly and independently. They then will show each other what they wrote and choose a restaurant according to the following rule: if the sum of the two numbers is 5 or less, they will go to a Mexican restaurant, if the sum is 7 they will go to an Italian restaurant and if the number is 9 or more they will go to a Japanese restaurant.

(a) Let Antonia be Player 1 and Bob Player 2. Represent this situation as a game frame, first by writing out each element of the quadruple of Definition 1.1 and then by using a table (label the rows with Antonia’s strategies and the columns with Bob’s strategies, so that we can think of Antonia as choosing the row and Bob as choosing the column).

(b) Suppose that Antonia and Bob have the following preferences (where $M$ stands for ‘Mexican’, I for ‘Italian’ and J for ‘Japanese). For Antonia: $M \succ_{\text {Antonia }} I \succ_{\text {Antania }} J$; for Bob: $I \succ_{\text {Bob }} M \succ_{\text {Bob }} J$. Using utility function with values 1,2 and 3 represent the corresponding reduced-form game as a table.

Exercise 1.2. Consider the following two-player game-frame where each player is given a set of cards and each card has a number on it. The players are Antonia (Player 1) and Bob (Player 2). Antonia’s cards have the following numbers (one number on each card): 2, 4 and 6, whereas Bob’s cards are marked 0, 1 and 2 (thus different numbers from the previous exercise). Antonia chooses one of hers own cards and Bob chooses one of his own cards: this is done without knowing the other player’s choice. The outcome depends on the sum of the points of the chosen cards, as follows. If the sum of points on the two chosen cards is greater than or equal to 5 , Antonia gets $\$ 10$ minus that sum; otherwise (that is, if the sum is less than 5) she gets nothing; furthermore, if the sum of points is an odd number, Bob gets as many dollars as that sum; if the sum of points turns out to be an even number and is less than or equal to 6 , Bob gets $\$ 2$; otherwise he gets nothing.

(a) Represent the game-frame described above by means of a table. As in the previous exercise, assign the rows to Antonia and the columns to Bob.

(b) Using the game-frame of part (a) obtain a reduced-form game by adding the information that each player is selfish and greedy. This means that each player only cares about how much money he/she gets and prefers more money to less.

Exercise 1.4. There are two players. Each player is given an unmarked envelope and asked to put in it either nothing or $\$ 300$ of his own money or $\$ 600$. A referee collects the envelopes, opens them, gathers all the money, then adds $50 \%$ of that amount (using his own money) and divides the total into two equal parts which he then distributes to the players.

(a) Represent this game frame with two alternative tables: the first table showing in each cell the amount of money distributed to Player 1 and the amount of money distributed to Player 2, the second table showing the change in wealth of each player (money received minus contribution).

(b) Suppose that Player 1 has some animosity towards the referee and ranks the outcomes in terms of how much money the referee loses (the more, the better), while Player 2 is selfish and greedy and ranks the outcomes in terms of her own net gain. Represent the corresponding game using a table.

(c) Is there a strict dominant-strategy equilibrium?

Exercise 1.7. For the second-price auction partially illustrated in Table 1.9, complete the representation by adding the payoffs of Player 2, assuming that Player 2 assigns a value of $\$ 50 \mathrm{M}$ to the field and, like Player 1, ranks the outcomes in terms of the net gain from the oil field (defined as profits minus the price paid, if Player 2 wins, and zero otherwise).