Finitely Repeated Games

First, an aside: The Federal Open Market Committee meets to formulate the implementation of monetary policy. The policy prescription is executed by the New York Fed, which enters the bond markets to buy and sell U.S. government obligations. At the same time the U.S. Treasury sells securities in order to finance federal expenditures and refinance maturing debt. In the fifties an accord was necessary to bring the activities of these two independent agencies into closer harmony.

When the Treasury sells government obligations it does so by auction. The purchasers are firms like Salomon, Merrill Lynch and Morgan Stanley, who turn around and sell the bonds to pension funds and individual investors. This activity was the basis for the book and movie titled "Bonfire of the Vanities."

A week before the auction the Treasury issues a press release announcing the auction date, the amount, and type of securities to be offered. The type can be 13-week, 26-week bills (offered weekly), two- and five year notes (offered monthly). If you are interested in buying then you submit a "tender" stating the amount you want to buy and the price you are willing to pay. Once everyone has submitted their bid the Treasury determines a fixed price that they will charge on a given lot of securities (a single price auction) or a range for the price (a multi-price auction). In a multi-price auction the highest bidder is allocated her demand first, then the next highest bidder gets the quantity he demanded, and so on.

This is a repeated game because it is played every week of the year. To simplify matters we will stipulate that the Treasury sells 100 bonds each week. There are only two bidders, Jill and Jack. The two buyers can offer to buy in one of two quantities (50 and 75), and one of two prices (high and low). Thus, each player has four moves: <50, high>, <50, low>, <75, high>, and <75, low>. The buyers are also interested only in profits. If they pay a high price for the bond then their profit on each bond is

. If they pay a low price for the bond then their profit per bond is

. Remember that the subscript refers to the price paid for the bond. Therefore, the bidder makes more profit when they pay a low price than when they pay a high price. That is,

If both players offer a high price then that is the market price and total demand is at least 100. If both players offer a low price then that is the market price and total demand is at least 100. Suppose one player offers a high price and the other offers a low price. In a single price auction the market price is set at the lower bid. In a multi-price auction the high bidder gets all of the bonds he asks for at that price and then the low price bidder gets the remaining bonds she wants at her offer price. Whenever the bid prices are the same then the quantity is allocated proportionately to the the quantity demanded. If Jill wants 75 bonds and Jack wants 50, then Jill gets 75/(50+75) = 60 and Jack gets the rest, 40.

Payoffs in the Single Price Auction
		Jack
		50, high	75, high	50, low	75, low
Jill	50, high	,	,	,	,
	75, high	,	,	,	,
	50, low	,	,	,	,
	75, low	,	,	,	,

The payoffs are read as follows: Price is determined by the lower of the two bids. The higher bidder gets his/her demand filled first and the rest goes to the low bidder. If the bid prices are the same then the 100 bonds are allocated in proportion to the quantity requested in the bid. For example, if Jack bids <75, high> and Jill bids <50, low>, then the price is set at 'low', Jack's request for 75 bonds is met since he is the high bidder and Jill gets the remaining 25 bonds.

We can reduce the size of this strategic game by elimination of dominated strategies. For either player <75, low> dominates <50, low>, and <75, high> dominates <50, high>. Therefore we need only look at the 2x2 strategic game.

In the single price game if

, then the dominant strategy for both players is to bid a high price and ask for 75 bonds. This makes the Treasury happy since high, high is the dominant strategy no matter how often the game is played. When the inequality is reversed then the dominant strategy for each player is a mixed strategy. When the players play mixed strategies then the Treasury may realize low proceeds from the sale, but at least they get high proceeds some of the time.

The payoff table for the multi-price auction is a little different. Remember that in the multi-price auction each participant pays the price bid, then the bonds are allocated to the high bidder first. The lower bidder receives whatever is left at the lower price. The table looks like

Notice that some of the payoffs differ from the single price auction since each bidder pays his own price. For Jack <75, low> dominates <50, low>, and <75, high> dominates <50, high>. For Jill <75, low> dominates <50, low>, and <75, high> dominates <50, high>. After elimination of the dominated strategies the payoffs are

Payoffs in the Multi-price Auction After Eliminating Dominated Strategies
		Jack
		75, high	75, low
Jill	75, high	,	,
Jill	75, low	,	,

Suppose that in this multi-price game the best response to a high bid is to bid high, as in the single price game. But suppose in addition, the best response to a low bid is to also bid low. That is

. Now there are two Nash equilibria. If the game is played repeatedly then the players will alternate high and low bids. Furthermore, if they can get away with it, the bidders have an incentive to collude and submit low bids!

Suppose that

, the reverse of the earlier case. In the multi-price case 'low' is now the dominant strategy. The Treasury doesn't like this since they realize smaller proceeds from the auction.

Payoffs in the Single Price Auction After Eliminating Dominated Strategies
		Jack
		75, high	75, low
Jill	75, high	,	,
Jill	75, low	,	,

Payoffs in the Multi-Price Auction
		Jack
		50, high	75, high	50, low	75, low
Jill	50, high	,	,	,	,
	75, high	,	,	,	,
	50, low	,	,	,	,
	75, low	,	,	,	,