Welcome to the Math 17 Web Page

Math 17

An Introduction to Mathematics Beyond Calculus:
The Many Faces of Game Theory

Instructor: Prof. Peter Winkler (peter.winkler at dartmouth.edu)

### Abstract

Game Theory is one of the great accomplishments of modern mathematics, with myriad applications to economics, computer science, political science, decision theory, warfare, and even---occasionally---actual games played for fun. This course will tackle the key results in each of the main branches of game theory, using a variety of mathematical techniques.

We will begin with the simplest and most famous game model: two-player, simultaneous, one-move games such as the "prisoners' dilemma," and the notion of Nash equilibrium. From this we will branch out to multiple players, repeated games, auctions and voting. Later we will digress to the fascinating world of combinatorial games, talking about value and strategy.

Prerequisites: Math 8, placement into Math 11, or permission of the instructor. Dist: QDS.

Here is a (tentative) rough weekly syllabus.

1. Matrix games; dominance, mixed strategies, expection

2. Equilibria and cooperation

3. Multiple players and coalitions

4. Equitability and stable matching

5. Auction design

6. Voting schemes and voting power

7. Repeated games

8. Combinatorial games

9. Alternating-turn and random-turn games

### Classes

Room: Kemeny Hall 201
Lectures: "10" slot, in particular: Monday, Wednesday and Friday 10-11:05.
X-hour: Thursdays (same room) 12:00 pm--12:50pm. Will be used only when so announced in class.

### Staff

Instructor:
Peter Winkler -- Kemeny Hall 231 / Tel. 6-3468
Office Hours: M 2:00-3:00; W 1:30-2:30; F 2:00-3:00.

### Textbooks

Steven Tadelis, Game Theory: An Introduction, Princeton University Press, 2012.

Your grade will be based on homework (20%), class participation (10%), two in-class exams (15% each) and final exam (40%).

### Exams

There will be two in-class hour exams, the first on Monday Oct 7, the second on Monday Oct 28.
Let me know immediately if you might not be able to make it to an exam.
The final exam will be in the regular slot for MWF10 courses, namely 8am(!) Friday, Nov 22.

### Homework

Homework will be assigned at each class period, due at the beginning of the next class.

### Assignments

Due Wednesday Sept 18: Read Chapter 1 of the text, and Chapter 2 too, if you like. Do Exercise 1.3, namely:

Fruit or Candy: A banana costs \$0.50 and a piece of candy costs \$0.25 at the local cafeteria. You have \$1.25 in your pocket and you value money. The money-equivalent value (payoff) you get from eating your first banana is \$1.20, and that of each additional banana is half the previous one (the second banana gives you a value of \$0.60, the third \$0.30, and so on). Similarly the payoff you get from your first piece of candy is \$0.40, and that of each additional piece is half the previous one (\$0.20, \$0.10, and so on). Your value from eating bananas is not affected by how many pieces of candy you eat, and vice-versa.

a. What is the set of possible actions you can take given your budget of \$1.25?

b. Draw the decision tree that is associated with this decision problem. (A decision tree has branches labeled by each successive decision; for example, in this problem you could start with a 3-way branching between starting with one banana, starting with one piece of candy, or keeping all your money.)

c. Would you spend all your money at the cafeteria? Justify your answer with a rational choice argument.

d. Now imagine that the price of a piece of candy increases to \$0.30. How many possible actions do you have? Does your answer to (c) change?

Due Friday Sept 20: Read Chapter 2, to make sure you've understood the class material, and start Chapter 3. Do Exercize 2.3 (dog races).

Due Monday Sept 23: Read Chapter 3, to make sure you've understood the class material, but skip 3.1.2 unless you're an economics major. You can read 4.1 if you want a head start on Monday's class. Do Exercize 3.2 (penalty kicks) and write it up to hand in in class.

Due Wednesday Sept 25: Read Chapter 4, Sections 1 and 2, to make sure you've understood the class material, but skip Carnot Duopoly. Write up Exercize 4.3 and look at 4.4 (eBay) if you like.

Due Friday Sept 27: Finish Chapter 4, and do Exercise 4.7 (Campaigning) to hand in Friday.

Due Wednesday Oct 2: (not to be handed in) Let Tn denote the tree-shaped Hackenbush game with n+1 edges: one blue "trunk" edge touching the ground, and n red "branch" edges that all meet at the upper vertex of the trunk. Note that we showed in class that the value of T1 is 1/2.

a) Argue that the value of Tn is strictly positive for all positive n.

b) Prove that the value of T2 is 1/4.

c) Prove that the value of T3 is 1/8.

Due Friday Oct 11: Read Chapter 5 and begin 6, skipping the duopoly stuff. To hand in: find a Nash equilibrium for Rock, Paper, Scissors where Alice's broken finger prevents her from playing "scissors".

Due Monday Oct 14: Finish reading Chapters 5 and 6 (minus duopoly stuff), and do Exercises 5.4 (pizza) and 6.5 (cop and robber) to to hand in in class.

Due Wednesday Oct 16: Do and write up Exercise 6.9 (all-pay auction), and begin reading Chapter 7 (7.1 and 7.2).

Due Friday Oct 18: Read Chapter 7, and hand in the following. Alice and Bob each ante \$1. A random card is dealt to Alice (with probability 1/4, it is a spade, but only Alice gets to see it.) Alice may fold (in which case Bob claims the pot, thereby winning Alice's \$1). Alternatively, she may bet \$10. If she does, Bob may fold or call. If he folds, Alice claims the pot with Bob's \$1 in it. Finally, if he calls, Alice shows her card and if it's a spade, she wins \$11 (\$1 ante + \$10 call) from Bob; otherwise it's Bob who wins \$11 from Alice.

(a) Make a normal-form matrix for this game. What is the Nash equilibrium? (b) To whose advantage is the game? (c) How, if possible, would you tip the game to the other player's advantage by substituting a different amount for \$10?

Due Monday Oct 21: For the women: Who wins tic-tac-toe when either player can play an X or an O, the winner being the first to achieve three X's or three O's in a row?

For the men: Who wins tic-tac-toe with an extra, tenth square extending the bottom row to the right? As normal, first player plays only X and second only O. To win on the bottom row, you need four of your symbol. But note that there is now an extra diagonal on which to win with three in a row.

Due Wednesday Oct 23: Determine (and hand in) whether it is the first or second player who has the advantage in turn-die, a game played as follows: A die is rolled, giving a number, say x1, between 1 and 6. Player 1 turns the die to give a number x2 (which cannot then be equal to x1 or 7 - x1), and the sum x1 + x2 is determined. Then Player 2 turns the die again, giving x3 (not equal to x2 or to 7 - x2), and the new sum x1 + x2 + x3 is calculated. The players continue to alternate until a sum of 21 or higher is reached. A player wins if either he hits the sum of 21 exactly or his opponent causes the sum to exceed 21. (Hint: a state of the game can be described by knowing the current sum and whether the die is now at 1 or 6, at 2 or 5, or at 3 or 4.)

Due Friday Oct 25: A deck of cards is spread face up on the table. Alice choses five for her hand, then Bob chooses five for his. Now Alice throws out some cards (any number from zero to all five) and replaces them with a like number from the deck. (The cards she discards go out of play.) Bob does likewise. The hands are compared by the usual poker rules, best being AKQJ10 of a suit, next best KQJ109 of a suit, etc; tie goes to Bob. Who wins, and how, with best play?

Due Wednesday Oct 30: Write up and hand in your solution to last Wednesday's poker problem, detailing all winning first-hands for Alice.

Due Friday Nov 1: Alice and Bob each choose (simultaneously) "a" or "b". If both choose the same letter, each earns \$1; if different, the one who chose "b" earns \$3 while the other earns nothing. Consider playing this game once, twice, or infinitely often. What can you say about Nash equilibria? Are there any where both players use the same (possibly mixed) strategy? Are there any with pareto-optimal outcomes? (You don't need to write this up.)

Due Monday Nov 4: (to write up and hand in) Alice and Bob each choose (simultaneously) "a" or "b". If both choose the same letter, each earns nothing. If different, the one who chose "b" earns \$2 while the other earns \$1. The game is repeated infinitely often, with discount factor .9. Find two symmetric Nash-equilibrium solutions each of which is Pareto-optimal.

Due Wednesday Nov 6: (to think about) I own a widget whose value to me is known to me, but as far as you know, that value is equally likely to be anything from \$0 to \$100. You do know, however, that because you're much better than me at operating the widget, its value to you is 80% higher than its value to me.

You get to make one bid of \$x. If \$x is more than the widget's value to me, I'll sell it to you for \$x. Otherwise, no sale. How much should you bid?

Due Friday Nov 8: (to write up and hand in) Alice and Bob are sole participants in a sealed-bid, first-price auction in which any multiple of \$100 is accepted as a bid, and in the event of a tie, a coin-flip determines who gets the item (and pays his/her bid for it). It is common knowledge that the item is worth \$500 to either player. Find all Nash equilibria for this game.

Due Monday Nov 11: (to think about) Alice, Bob and Carla need to determine whether a certain coin is heads-biased (2/3 heads, 1/3 tails) or the reverse. Each flips the coin once, privately, and then votes "heads" or "tails"; the vote may not correspond to the flip. If all three vote "tails" the committee is deemed to have decided that the coin was tails-biased, otherwise heads-biased. If the committee is right, everyone earns \$1, otherwise nothing. What are the Nash-equilibrium solutions to this game?

Due Wednesday Nov 13: (to write up and hand in) Compute the optimal (biggest expected revenue) reserve price to place on a second-price sealed-bid auction (or, equivalently, an English auction) whose two bidders have independent valuations that are uniformly random in [0,1]. Assume the object in question will be destroyed if it is not sold. (For 1 point extra credit, do it for n bidders instead of just two.)

Due Friday Nov 15: (to think about) As in Monday's assignment, Alice, Bob and Carla need to determine whether a certain coin is heads-biased (2/3 heads, 1/3 tails) or the reverse. Again, each flips the coin once privately and then votes "H" or "T". This time, there is 0 payoff if the three of them get it right; -1 if they conclude "heads" when the coin is tails-biased; and -x if they conclude "tails" when the coin is heads-biased. For what range of values for x is Monday's "mechanism"---that is, concluding "heads" unless all three votes are "T"---the best scheme?

### Honor Code

Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. The honor principle applies to homework in the following way. What a student turns in as a written homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. Students are discouraged from using solutions to problems that may be posted on the web, and as just stated, must reference them if they use them. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me, and I will be glad to help clarify things. It is always easier to ask beforehand than to have trouble later!

### Disabilities

I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

The Student Disabilities Center is located at 318 Wilson Hall, ext. 6-9900, http://www.dartmouth.edu/~accessibility, if you have any questions. Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.