game theory and auctions - research ict africa · what is game theory? mathematically capture...

Game Theory and AuctionsDr Christoph Stork

Wednesday, 8 August 12

What is Game Theory?Mathematically capture behaviour in strategic situations (games) in which an individual's success in making choices depends on the choices of othersPsychologists call the theory of social situationsEconomist: Game theoryDominance: when one strategy is better than another strategy for one player, no matter how that player's opponents may playIntransitivity: one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play


http://en.wikipedia.org/wiki/Strategy_(game_theory)

http://en.wikipedia.org/wiki/Strategy_(game_theory)

http://en.wikipedia.org/wiki/Intransitivity

http://en.wikipedia.org/wiki/Intransitivity

ApplicationsAuctions (spectrum, licences, art, cattle)Wage bargaining: trade unions vs employersPrice setting in oligopoliesContract negotiationsTendering for government projectsVoting in ParliamentCorruption in institutions


Prisoner's DilemmaTwo partners in a crime who have been captured by the police. The police does not have evidence against them but both were carrying gunsEach suspect is placed in a separate cell, and offered the opportunity to confess to the crime.If neither suspect confesses, they will only be charged is illegal possession of fire arms and be jailed for a year. If one prisoner confesses and the other does not, the prisoner who confesses testifies against the other in exchange for going free, the other is jailed for 10 yearsIf both prisoners confess, then both are given a reduced term, but both are convicted for 5 years in jail.


Not Confess Confess

Not Confess

Confess

1 1 10 0

0 10 5 5


Prisoner's DilemmaNo matter what a suspect believes his partner is going to do, it is always best to confess: Dominant Strategy If the partner in the other cell is not confessing, it is possible to go free.If the partner in the other cell is confessing, it is possible to get 5 years instead of 10 years in prisonYet if neither confessed, both would only get 1 year, which for both together would be the best.This conflict between the pursuit of individual goals and the common good is at the heart of many game theoretic problems.


Changes to GameWhat if Game is repeated?Players can talk to each other?


Two- person Zero-sum game

Stretch of beach with eighty sunbathers evenly distributed along it. 2 ice-cream sellers: you are oneBest positions for you and the other ice-cream seller are A and B (minimise the average distance any sunbather has to walk for an ice-creamStarting at position A, and that the other starting at B, You sell to the region LAC, and the other seller sells to the region CBRby moving a little to the right you can increase your expected sales by capturing some of the other seller’s marketother seller will realise that by moving a little to the left he can recapture those of his previous customers to the right of CShuffling continuing until each seller is at the centre of the beach, with equal sales

1. On the Beach . . .1.1 A Hot Afternoon

CONSIDER a stretch of beach with eighty sunbathers evenly distributed alongit. There are two ice-cream sellers: you are one. As far as the sunbathers are

concerned, the best positions for you and the other ice-cream seller are A and B inthe figure, the quarter and three-quarters positions along the beach, since theseminimize the average distance any sunbather has to walk for an ice-cream, one-eighth of the beach. Suppose you are the left-hand seller starting at position A, andthat the other seller starts at position B in the figure. You now sell to the regionLAC, and the other seller sells to the region CBR.

| | | | |L A C B R

Since the sunbathers are evenly distributed between L and R, and are otherwiseidentical, you and the other seller each expect to sell an equal number, forty ice-creams.

But now you realise that by moving a little to the right you can increase yourexpected sales by capturing some of the other seller’s market, those sunbathers justto the right of C, who now find it more convenient (shorter) to walk to the left fortheir ice-creams than to the right, as previously. At the same time, of course, theaverage distance your customers have to walk increases, since you are farther tothe right. But the other seller will not be content to lose his market: he will realisethat by moving a little to the left he can recapture those of his previous customersto the right of C who now find it more convenient to buy from you. Indeed, hemight well move closer to C than you are, in order to capture some of the marketon the left half of the beach.

We can imagine this shuffling continuing until each seller is at the centre ofthe beach, with equal sales. If we ignore the possibility of sunbathers near L and Rdeciding that the extra distance to the ice-cream is excessive, neither seller willhave gained or lost any new customers. We speak of this situation as a “two-person, zero-sum game,” since the total market is unchanged: there is directconflict, with one seller’s gain being the other seller’s loss. It is possible to speakof two alternatives facing each seller: move to the centre (M) or remain at A or B(NM). We can then write two “payoff matrices,” showing values to each seller ofthe four combinations of strategies:

The values shown in Table 1 are your sales, given the actions of you and theother seller: if neither seller moves, you have half the market (from L to C); this isalso the case if both sellers move (to the centre), assuming that no customers arediscouraged by the doubling of the average distance to the ice-cream; if you moveto the centre while the other seller remains at B, then your sales increase (in thelimit to fifty with perfect knowledge, no customer loyalty, and no customerdiscouragement); if you remain at A while the other seller moves to the centre,

- 2 -


Move Not Move

Move

Not Move

40 40 50 30

30 50 40 40


What happens if bathers are lazy?

1. On the Beach . . .1.1 A Hot Afternoon

CONSIDER a stretch of beach with eighty sunbathers evenly distributed alongit. There are two ice-cream sellers: you are one. As far as the sunbathers are

concerned, the best positions for you and the other ice-cream seller are A and B inthe figure, the quarter and three-quarters positions along the beach, since theseminimize the average distance any sunbather has to walk for an ice-cream, one-eighth of the beach. Suppose you are the left-hand seller starting at position A, andthat the other seller starts at position B in the figure. You now sell to the regionLAC, and the other seller sells to the region CBR.

| | | | |L A C B R

Since the sunbathers are evenly distributed between L and R, and are otherwiseidentical, you and the other seller each expect to sell an equal number, forty ice-creams.

But now you realise that by moving a little to the right you can increase yourexpected sales by capturing some of the other seller’s market, those sunbathers justto the right of C, who now find it more convenient (shorter) to walk to the left fortheir ice-creams than to the right, as previously. At the same time, of course, theaverage distance your customers have to walk increases, since you are farther tothe right. But the other seller will not be content to lose his market: he will realisethat by moving a little to the left he can recapture those of his previous customersto the right of C who now find it more convenient to buy from you. Indeed, hemight well move closer to C than you are, in order to capture some of the marketon the left half of the beach.

We can imagine this shuffling continuing until each seller is at the centre ofthe beach, with equal sales. If we ignore the possibility of sunbathers near L and Rdeciding that the extra distance to the ice-cream is excessive, neither seller willhave gained or lost any new customers. We speak of this situation as a “two-person, zero-sum game,” since the total market is unchanged: there is directconflict, with one seller’s gain being the other seller’s loss. It is possible to speakof two alternatives facing each seller: move to the centre (M) or remain at A or B(NM). We can then write two “payoff matrices,” showing values to each seller ofthe four combinations of strategies:

The values shown in Table 1 are your sales, given the actions of you and theother seller: if neither seller moves, you have half the market (from L to C); this isalso the case if both sellers move (to the centre), assuming that no customers arediscouraged by the doubling of the average distance to the ice-cream; if you moveto the centre while the other seller remains at B, then your sales increase (in thelimit to fifty with perfect knowledge, no customer loyalty, and no customerdiscouragement); if you remain at A while the other seller moves to the centre,

- 2 -

Me The other guys


Lazy Bathers Move Not Move

Move

Not Move

20 20 40 30

30 40 40 40


Bar GAMEBar GAMEBar GAMEBar GAMEZeno LoungeZeno Lounge

Promotion No Promotion

El Cubano

Promotion

El Cubano

No Promotion

10 14 18 6

4 20 7 8


2011/01/23 1:59 PMGame Theory

Page 4 of 6http://www.quickmba.com/econ/micro/gametheory/

Structure of an Extensive Form Game

Chance nodes can appear anywhere in the extensive form tree.

An information set is a collection of nodes that are controlled by the same player,but which are indistinguishable for that player. In other words, for nodes in the sameinformation set, the player does not know which one he/she is at, but does knowthat these nodes are different. In the preceding diagram, the drawn information setsmight arise if Decision A and Decision C were indistinguishable to Player 2, as wellas Decision B and Decision D. If a single dotted line encompassed all the Player 2decision nodes (or 4 dotted circles all connected), then Player 2 would not be ableto distinguish between any of the four decisions.

An extensive form game without information sets designated is one in which theplayers know exactly where they are in the tree. This situation is equivalent to oneof dotted circles drawn around each decision point in the tree but not connected toone another. If neither player can observe anything about the other player's action,the sequential extensive form game can be reduced to the simultaneous-actionbimatrix game.

Normal-Form (Strategic Form) Game Representation

The extensive form of representing a game can become difficult to manage as thegame gets larger, and the Nash equilibria may become difficult to find. Theextensive form representation can be collapsed into the normal form, which encodesthe game into a strategy that describes the action to take for each conceivablesituation (for example, for each information set). The normal form is a complete


Nash EquilibriumA combination of player,s strategies that are best responses to each other?Examples:Going out with matesInternational Trade agreements


Types of Auctions


Exercise: LTE Spectrum Auction

3*10 Mhz in 2.2 GHZ band: Consignment A, B and C, each having 2*5 MhzYou receive a piece of paper with your company name, the value the spectrum is worth to your business and the maximum financial resources you haveYou will be discussing your bidding strategy with your business partners before biddingThe winner and the final amount will be announced after each roundYour personal bonus will be Economic value for you business minus accepted bid price5 companies are allowed to bid: MTC, Leo, Telecom Namibia, ITN, Africa Online


http://en.wikipedia.org/wiki/Winner%27s_curse

http://en.wikipedia.org/wiki/Winner%27s_curse

Open Outcry AuctionBidders are in the same room and indicate the price they are willing to pay as it is called in an ascending manner or descending mannerThe winner is the one that offers the highest priceThe winner could either pay the prize he/she indicated or can be offered to pay the nearest lower price depending on the auction designWhen the bidder pays the price he bid for it is called the first price auction and when he is offered to pay the next higher price, which is lower than the winning bid, it is called the second lowest price bid or second price auction


Dutch auctionAuctioneer begins with a high asking price which is lowered until some participant is willing to accept the auctioneer's price, or a predetermined reserve price (the seller's minimum acceptable price) is reachedThe winning participant pays the last announced price. Also known as: Clock auction or open-outcry descending-price auctionthe bidding strategy and results of this auction are equivalent to those in a sealed first-price auction. Dutch auction is named for its use in the Dutch Tulip Craze


http://en.wikipedia.org/wiki/Asking_price

http://en.wikipedia.org/wiki/Asking_price

http://en.wikipedia.org/wiki/Reservation_price

http://en.wikipedia.org/wiki/Reservation_price

http://en.wikipedia.org/wiki/Sealed_first-price_auction

http://en.wikipedia.org/wiki/Sealed_first-price_auction

http://en.wikipedia.org/wiki/Dutch_Tulip_Craze

http://en.wikipedia.org/wiki/Dutch_Tulip_Craze

Single Sealed bidBidders submit the highest price they are willing and able to pay in a sealed envelopeRegulator opens all bids in the presence of bidders and reads out the bidsHighest bidder is the winnerDepending on the design of the auction the winner could pay the highest price they bid for or the second highest bid

First-price sealed-bid auction (Used at the London Gold Exchange)Second-price sealed-bid auctions (Vickrey auctions)

Bidders reveal their true value for the licence Paying a lower price than they bid for could reward the winner


English auction:Open Ascending-bid auctions

A type of sequential second price auction in which an auctioneer directs participants to beat the current, standing bid (by increment)New bids must increase the current bid by a predefined incrementThe auction ends when no participant is willing to outbid the current standing bidParticipant who placed the current bid is the winner and pays the amount bidWinning bidder needs only to outbid the next highest bidder by the minimum increment. Thus the winner, effectively, pays an amount equal to (slightly higher than) the second highest bid


Combinatorial/ Packaged Auction

Bidders have an interest in winning various licences that are complementary Bidder is allowed to bid for a combination of licences he wants or for each individual one separatelyThe bidder will tend to offer the true value reflection of the licences in an open simultaneous or sequential auctionHowever, it is complex to run and participate in a simultaneous package than sequential packaged auctions because the choices could be too many and confusing


Multi-Unit AuctionSeveral items are sold: Each at the same price (a uniform price auction) or at different prices (a discriminatory price auction)Uniform Auction

fixed number of identical units of a commodity are sold for the same priceEach bidder in the auction bids a price and a quantityThe price bid is considered the maximum price they are willing to pay per item, and the quantity is the number of units they wish to purchase at that priceTypically these bids are sealed - not revealed to the other buyers until the auction closesThe auctioneer then serves the highest bidder first, giving them the number of units requested, then the second highest bidder and so forth until the supply of the commodity is exhausted. All bidders then pay a per unit price equal to the lowest winning bid (the lowest bid out of the buyers who actually received one or more units of the commodity) - regardless of their actual bid.Some variations of this auction have the winners paying the highest losing bid rather than the lowest winning bid

Discriminatory auction: Highest bidder is the winner but the winners might have bid different prices for each unit


http://en.wikipedia.org/wiki/Uniform_price_auction

http://en.wikipedia.org/wiki/Uniform_price_auction

http://en.wikipedia.org/wiki/Discriminatory_price_auction

http://en.wikipedia.org/wiki/Discriminatory_price_auction

Simultaneous Ascending Auction

Most common form of auction used to allocate spectrum licences globallyBid for each item separately in each round The price goes up in each roundThe bids are sealed but announced at the beginning of each roundReserve price is placed on bidding on an item, bid must be more than the last bid by some determined marginAuction continues to take place until there are no new bids on any of the items There are penalties and restriction on how many items to bid for and for withdrawing or remaining inactive in the auctionThe winners are the highest bidders for a particular itemThis type of auction is regarded as transparent, fair, revenue maximising and efficient in determining the market value of the licence


game theory and auctions - research ict africa · what is game theory? mathematically capture...

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