Introduction to Auction Theory: Or How it Sometimes

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Archibald Tucker
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1 Introduction to Auction Theory: Or How it Sometimes Pays to Lose Yichuan Wang March 7, 20 Motivation: Get students to think about counter intuitive results in auctions Supplies: Dice (ideally per student) Battleship like papers Underlined things refers to questions that I want to make sure to ask the students so there s lots of interactions What is an auction? It s a way of selling things! Two key parts:. An object what people are trying to get 2. Bids what you re willing to pay Then an auctioneer must set rules that determine who gets the object depending on the bids, and how much to make the person pay Very common way of selling things. Flying 2. Paintings 3. Houses Common Value Auctions ( Hour) Pair off! You and a friend are going to play an auction game. Each person rolls a die, do not reveal it to the other person 2. Write down a bid for the sum of your dice + the other person s dice. Do not reveal the bid to the other person. 3. After writing it down, you reveal the bid by turning the paper around. 4. Whoever has the highest bid wins the auction, and then has a score of (Sum of Two Dice) - (The highest bid). If you lose the auction your score is 0. (a) Neither player wins if there is a draw Note you can any real number, so you can bid something like 3.5 or 2.25 or some other decimal

2 introduction to auction theory: or how it sometimes pays to lose 2 Do a demonstration Talk to your neighbors. What do you think is a good strategy for this game? Wait for some kid to mention something related to 3.5, or 7. Ask him why? Additional questions to prod: * * If you roll a high number do you think you should bid higher or lower? If you roll a higher number, do you think you re more likely to Let the kids play around 5-0 rounds of this. Go around to make sure that people are playing this correctly. What strategies did you try? For the kids who talked about just the average, see if that worked for them Let s explore some of the math The Idea of Expected Value Expected Value = Probability of Event Payoff from Event Events In the case of a dice, we can write out a tree Make sure this tree is still around

3 introduction to auction theory: or how it sometimes pays to lose 3 Payoff Probability Multiply You might have learned this as taking an average. Very important because it s actually a theorem that if you keep on rolling a dice, and take the average of the numbers that come up, you will eventually get very close to 3.5. More generally, if there s some random number, that keeps on appearing, as long as the probabilities stay the same, then the average gets close to this expected value Can also do a shortcut if each of the scenarios are equally likely. Can simplify as (Sum of Payoffs from Each Possibility) Number of Possibilities We can double check this for the dice, ( ) = 2 = 7 2 If this is the expected value for one dice, then what s the expected value for two dice? Expect at least one kid to say 7 Just say that if you have one random number, and then another random number, then the expected value of the sum is equal to the sum of the expected values

4 introduction to auction theory: or how it sometimes pays to lose 4 Intuition, if you were to roll both of the dice a billion times, and take an average, that should be approximately equal to the expected value But you should get literally the same number, if instead of adding and then averaging, you instead averaged and then added Ok, does it make sense to just always bid 7? What s the expected value of the sum if you already know that one of the dice is 5? Conditional Expected Value? No, because you already know what your dice says! You have information, and therefore you can condition on it So if you know you have a 5, then what is the expected value of the sum? Redo the tree diagram So the conditional expected value of the sum is your dice roll Given what you know about your own dice roll, you can have a better guess at what the sum is! Ok one last test, let s put this all together. Suppose we were rolling three dice. If you knew one of them was a 2, then what is you conditional expected value of the sum of the three dice? Let s play some more games! [Natural place to get some snacks] The Conceit of Expected Value Did anybody try this expected value strategy? Can you tell me your score on the rounds that you won the auction? Did people try different strategies? It turns out that the +3.5 strategy is pretty bad. One test: if both players play this strategy, what s the expected value? Formal proof is pretty annoying, Let s do a very concrete case. Suppose you roll a 4 and therefore bid 7.5. Suppose your opponent is following the same strategy

5 introduction to auction theory: or how it sometimes pays to lose 5 Opponent Opponent s Bid Winning Bid? Payoff = = = Draw Let s figure out the expected value from our bid # of Possibilities (Sum of Payoffs) = ( ) 0.75 You re better off not bidding at all! The problem: What is your expected value if you know that you re going to win the auction? If you win the auction, your bid is higher Which means your dice roll is higher than the other player Introduce concept of Nash Equilibrium Want to find a strategy such that if both players play the strategy, neither player has a reason to play any different strategy By this, it s clear that the +3.5 strategy is not Nash! You would rather just not play at all! Do you think you should bid more than your own dice? How much more?

6 introduction to auction theory: or how it sometimes pays to lose If there is time / students are at a sufficiently high skill level, can ask students to propose strategies, and then write the table of payoffs, where the rows index your roll, the columns the other player s roll, and then in each cell you have the payoff under that scenario. Then go around to each student, ask what expected value they computed, and then if their strategy is not pay your own bid, offer a suggestion. If they propose rule of (my dice +b), then typically + (b ɛ) is a better strategy. For more esoteric ones, ask what they would do if the other player were playing this strategy. What is the Optimal Strategy? Let s try to find a strategy so that if both players are playing it, then neither player has an incentive to change strategies. Called a Nash Equilibrium. Assumptions:. So let s start out by assuming that your opponent will bid whatever shows up on her dice. 2. Let s make another assumption. No ties. So in this rolling a sided dice game, it s actually a little bit hard to get an optimal strategy, but let s do a slight simplification. Let s pretend that you re playing with 00 sided dice and so it s really unlikely that the two of you get the same number.

7 introduction to auction theory: or how it sometimes pays to lose 7 Then you roll d. And you bid b. What s your payoff? Opponent Dice >b 0 Opponent Dice <b Opp. = 2 Opp. = 2. Opp. =b b First, what s the probability that you win the auction: Then what do you expect to win if you win the auction? b 00 In other words, what s the Expected Value of (Their Dice + Your Dice - Your Bid)? Well, remember that we can just find the expected value of their dice, your dice, and then add them. You already know what your dice is, it s just d. And your bid is just b. Let s figure out what the other expected value is. It must be the case that the other player rolled a number < b, and that each number < b is equally likely. So naturally the expected value is just the middle of all those numbers, which is 2 b. Hence the expected value is b 2 + d b Then let s multiply the probability you win the auction by the probability that you win the auction. That s the expected value. ( b d b ) 00 2 But wait, this looks just like a parabola! Two roots at 0 and 2d. Therefore the max is at d! Slightly complicated when things are discrete, but intuition is similar. Just gets complicated because you need to do rounding, etc.

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