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### Trade and Investment Strategy45-871

### Lecture 1 Auctions

Professor Robert A. Miller

Preliminaries

Before the lecture starts, if you haven’t already done so, please open your computer and go to:

http://comlabgames.com/

Then click on the comlabgames icon to reach:

http://www.comlabgames.com/free0.4/index.html

Download the stand alone version of comlabgames, which we will use throughout the course.

Please bring your laptop to each class.

Dynamic rent seeking

The first part of Trade and Investment Strategy is about exploring the market borders of your company through the greater use of internal markets. First we show how auctions work, and how they might be used in acquisition. Then we broaden the analysis to limit order markets, possibly internal, with multiple buyers and sellers.

The second part of Trade and Investment Strategyevaluates the dynamics of rent accrual, by linking firm net revenue flow to the pricing of financial securities and the consumption of goods and services. We divide the discussion into two parts, portfolio strategies for life-cycle saving and withdrawals, and direct investments as a high wealth management strategy.

Course objectives

In this course I hope you:

Become familiar with auctions and limit order markets as a bidder, trader, investor and analyst.

Acquire an overarching view for predicting and evaluating how trade and investment strategies are (or should be) linked to personal wealth management.

Develop an intuition for thinking about trading mechanisms and market based platforms as organizational tools.

Course materials

- The course website is:

http://www.comlabgames.com/45-871

- At the website you can find:
- the course syllabus
- lecture notes
- games you can download
- the on line (draft) textbook
- other reference materials

Auctions serve a vital role in business and they are relatively simple to analyze. That is why we begin the course with a study of auctions, and get some hands-on bidding experience. There are different types of auctions. We investigate how to bid, and then taking the auctioneers perspective, we ask what kind of auction produces the most revenue.

Auctions are an important trading mechanism

- Auctions are widely used by companies, private individuals and government agencies to buy and sell goods and services.
- They are also used in competitive contracting between an (auctioneer) firm and other (bidder) firms up or down the supply chain to reach trading agreements.
- Merger and acquisitions often have an auction flavor about them.

There are different types of auctions

- In a first price sealed bid auction, each bidder submits his/her bid without knowing what the others are bidding, and the auctioneer sells the good to the highest bidder at the price he submitted.
- In an English auction bidders compete against each other by raising the price until everyone but one bidder drops out of the bidding.
- In a Dutch auction, the auctioneer reduces the price until a bidder indicates he/she is willing to take the object.
- In a second priced sealed bid auction, players simultaneously submit their bids, the highest bidder wins the auction, and pays the second highest bid.

Why analyze auctions?

- Like all trading mechanisms, auctions serve the dual purpose of eliciting preferences and allocating resources between competing uses.
- Since an auction is the simplest form of a limit order market, starting this course by studying behavior in auctions is a useful way to begin learning how investors trade in any market.
- From a strategic perspective we seek answers to two basic questions:
- How should you bid in an auction?
- What kind of auction rules should you set?

Bidding strategies

- Does it matter what form the auction takes?
- From SCM (45-870) a strategy is a complete description of instructions to be played throughout the game
- The strategic form of a game is the set of alternative strategies to each player and their corresponding expected payoffs from following them.
- Two games are strategically equivalent if they share the same strategic form.
- In strategically equivalent auctions, the set of bidding strategies that each potential bidders receive, and the mapping to the bidder’s payoffs, are the same.

Common value auction:Oil field tract

- Consider a new oil field tract that drillers bid for after conducting seismic their individual explorations.
- The value of the oil field is the same to each bidder, but unknown. The nth bidder receives a signal sn which is distributed about the common value v, where

sn = v + n

and n E[v| sn] – v is independently distributed across bidders.

- Notice that each drilling company would have more precise estimates of the common valuation from reviewing the geological survey results of their rivals.

The expected value of the item upon winning the auction

- If the nth bidder wins the auction, he realizes his signal exceeded the signals of everybody else, that is

sn ≡ max{s₁,…,sN}

so he should condition the expected value of the item on this new information.

- His expected value is now the expected value of vn conditional upon observing the maximum signal:

E[vn| sn ≡ max{s₁,…,sN}]

- This is the value that the bidder should use in the auction, because he should recognize that unless his signal is the maximum he will receive a payoff of zero.

The Winner’s Curse

- Conditional on the signal, but before the bidding starts, the expectation of the common value is:
- We define the winner’s curse as:
- Although bidders should make due allowance for the fact that their valuation will typically overstate the true value of the object if they win the auction, novice bidders typically do not take it into account when placing a bid.

Descending auctions are strategically equivalent to first-price auctions

- During the course of a descending auction no information is received by bidders.
- Each bidder sets his reservation price before the auction, and submits a market order to buy if and when the limit auctioneer's limit order to sell falls to that point.
- Dutch auctions and first price sealed bid auctions share strategic form, and hence yield the same realized payoffs if the initial valuation draws are the same.

Rule 1: Pick the same reservation price in Dutch auction that you would submit in a first price auction

Second-price versus ascending auctions

- When there are only 2 bidders, an ascending auction mechanism is strategically equivalent to the second price sealed bid auction (because no information is received during the auction).
- More generally, both auctions are (almost) strategically equivalent if all bidders have independently distributed valuations (because the information conveyed by the other bidders has no effect on a bidder’s valuation).
- In common value auctions the two mechanisms are not strategically equivalent if there are more than 2 players.

Rule 2: If there are only two bidders, or if valuations are independently distributed, choose the same reservation price in English and second price auctions.

Bidding in a second-price auction

- If you know your own valuation, there is a general result about how to bid in a second price sealed bid auction, or where to stop bidding in an ascending auction.
- Bidding should not depend on what you know about the valuations of the other players, nor on what they know about their own valuations.
- It is a dominant strategy to bid your own valuation.
- A corollary of this result is that if every bidder knows his own valuation, then the object will be sold for the second highest valuation.

Rule 3 : In a second price sealed bid auction, bid your valuation if you know it.

Proving the third rule

- Suppose you bid above your valuation, win the auction, and the second highest bid also exceeds your valuation. In this case you make a loss. If you had bid your valuation then you would not have won the auction in this case. In every other case your winnings would have been identical. Therefore bidding your valuation dominates bidding above it.
- Suppose you bid below your valuation, and the winning bidder places a bid between your bid and your valuation. If you had bid your valuation, you would have won the auction and profited. In every other case your winnings would have been identical. Therefore bidding your valuation dominates bidding below it.
- The proof is completed by combining the two parts.

Revenue equivalence defined

- In strategically equivalent auctions, the strategic form solution strategies of the bidders, and the payoffs to all them, are identical. Are bidders ever indifferent to auctions that lack strategic equivalence?
- Two auction mechanisms are revenue equivalent if, given a set of players their valuations, and their information sets, the expected surplus to each bidder and the expected revenue to the auctioneer is the same.
- Revenue equivalence is a less stringent condition than strategic equivalence. Thus two strategic equivalent auctions are invariably revenue equivalent, but not all revenue equivalent auctions are strategic equivalent.

Preferences and Expected Payoffs

- Let P(vn) denote the probability the nth bidder with valuation vn will win the auction when all players bid according to their equilibrium strategy.
- Let C(vn) denote the expected costs (including any fees to enter the auction, and payments in the case of submitting a winning bid).
- Let:

U(vn) = P(vn) vn - C(vn)

denote the expected value of the nth bidder from following his equilibrium strategy when everyone else does too.

A revealed preference argument

- Suppose the valuation of n is vn and the valuation of j is vj.
- The surplus from n bidding as if his valuation is vj is U(vj), the value from participating if his valuation is vj, plus the difference in how he values the expected winnings compared to a bidder with valuation vj, or (vn – vj)P(vj).
- In equilibrium the value of n following his solution strategy is at least as profitable as deviating from it by pretending his valuation is vj. Therefore:

U(vn) > U(vj) + (vn – vj)P(vj)

Revealed preference continued

- For convenience, we rewrite the last slide on the previous page as:

U(vn) - U(vj) > (vn – vj)P(vj)

- Now viewing the problem from the jth bidder’s perspective we see that by symmetry:

U(vj) > U(vn) + (vj – vn)P(vn)

which can be expressed as:

(vn– vj)P(vn) > U(vn) - U(vj)

A fundamental equality

- Putting the two inequalities together, we obtain:

(vn – vj) P(vn)> U(vn) - U(vj) > (vn – vj) P(vj)

- Writing:

vn = vj + dv

yields:

which, upon integration, yields:

Revenue equivalence theorem

- This equality shows that in private value auctions, the expected surplus to each bidder does not depend on the auction mechanism itself providing the following conditions are satisfied:
- Every bidder is risk-neutral.
- Valuations are independent and identically distributed.
- In equilibrium the bidder with highest valuation wins.
- The lowest possible valuation has zero expected value.
- Note that if all bidders obtain the same expected surplus, the auctioneer obtains the same expected revenue too.

Intuition from revenue equivalence

- Calibrate your bid to your valuation only to the extent that it affects your beliefs about the highest valuation of the all the other bids.
- Working from the assumption that yours is the highest valuation, bid high enough to induce the next highest bidder to make a small expected loss in order to beat your bid.
- To use an athletic analogy, think heats, not finals! Swim fast enough to make the finals, but save yourself for the final.

Steps for deriving expected revenue

- The expected revenue from any auction satisfying the conditions of the theorem, is the expected value of the second highest bidder.
- To obtain this quantity, we proceed in two steps:

1. derive the probability distribution of the second highest valuation

2. obtain its density and integrate to find the mean.

Probability distribution of the second highest valuation

- Since any auction satisfying the conditions for the theorem can be used to calculate the expected revenue, we select the second price auction.
- The probability that the second highest valuation is less than v is the sum of the the probabilities that:

1. all the valuations are less than v, or P(v)N

2. N-1 valuations are less than v and the other one is greater than v. There are N ways of doing this so the probability is: NP(v)N-1[1 - P(v)]

- The probability distribution for the second highest valuation is therefore: NP(v)N-1 - (N - 1) P(v)N

Expected revenue from Private Value Auctions

- The probability density function for the second highest valuation v is therefore:

N(N –1)P(v)N-2 [1 - P(v)]P’(v)

- Therefore the expected revenue to the auctioneer, or the expected value of the second highest valuation, denoted by , is:

Using the revenue equivalence theorem to derive optimal bidding functions

- We can also derive the solution bidding strategies for auctions that are revenue equivalent to the second price sealed bid auction.
- Consider, for example a first price sealed bid auctions with independent and identically distributed valuations.
- The revenue equivalence theorem implies that each bidder will bid the expected value of the next highest bidder conditional upon his valuation being the highest.

Bidding in a first price sealed bid auction

- In a symmetric equilibrium to first price sealed bid auction, we can show that a bidder with valuation vn bids:

Comparison of bidding strategies

- The bidding strategies in the first and second price auctions markedly differ.
- In a second price auction bidders should submit their valuation regardless of the number of players bidding on the object.
- In the first price auction bidders should shave their valuations, by an amount depending on the number of bidders.

The derivation

- The probability the remaining N - 1 valuations are less than v given the highest valuation is vn is:
- Differentiating, the conditional density for the second highest valuation is then:
- If he wins, the bidding function for n is the expected value of the second highest valuation:
- Integrating by parts, we simplify this formula to:

An example: the uniform distribution

- Suppose valuations are uniformly distributed within a closed interval, with probability distribution:
- Then in equilibrium, a player with valuation v bids a weighted average of the lowest possible valuation and his own, where the weights are 1/N and (N-1)/N:

Choosing between two auctions with revenue equivalence

- Since first and second price auctions are revenue equivalent, why would one choose one over the other?
- The relationship winning bid and the price paid is more transparent in a first price sealed bid than in the second price sealed bid.
- In the first price sealed bid you pay what you bid, rather than what the auctioneer claims is the second highest bid.
- On the other hand the English auction (strategically equivalent in private value auctions) also is transparent because bidders see their rivals.

Preventing collusion

- One reason for the auctioneer to prefer a first price auction is that that is easier to prevent collusion under a first price auction than under a second.
- In a second price auction a firm deviating from the cartel agreement can be easily penalized by the cartel members by the designated winner offering a bid that would ensure a loss.
- In a first price auction the designated winner makes a bid that has attractive terms for itself, so the deviating firm could still receive a profitable contract by making a slightly better contract.

What happens when more than one unit is sold?

- Suppose there are exactly Q identical units of a good up for auction, all of which must be sold.
- As before we shall suppose there are N bidders or potential demanders of the product and that N > Q.
- Also following previous notation, denote their valuations by v1 through vN.
- We begin by considering situations where each buyer wishes to purchase at most one unit of the good.

Open auctions for selling identical units

- Descending Dutch auction:

As the price falls, the first Q bidders to submit market orders purchase a unit of the good at the price the auctioneer offered to them.

- Ascending Japanese auction:

The auctioneer holds an ascending auction and awards the objects to the Q highest bidders at the price the N - Q highest bidder drops out.

Multiunit sealed bid auctions

- Sealed bid auctions for multiple units can be conducted by inviting bidders to submit limit order offers, and allocating the available units to the highest bidders.
- In discriminatory auctions the winning bidders pay different prices. For example they might pay at the respective prices they posted.
- In a uniform price auction the winners pay the same price, such as a kth price auction (where k could range from 1 to N.)

Revenue equivalence revisited

- Suppose each bidder:

- knows her own valuation

- only want one of the identical items up for auction

- is risk neutral

- Consider two auctions which both award the auctioned items to the highest valuation bidders in equilibrium.
- Then the revenue equivalence theorem applies, implying that the mechanism chosen for trading is immaterial.
- Intuitively each bidder tries to beat the highest losing bid.

Successful bids follow a random walk

- In repeated auctions that satisfy the revenue equivalence theorem, EMH implies that the price of successive units follows a random walk.
- Intuitively, each bidder is estimating the bid he must make to beat the demander with (Q+1)st highest valuation, that is conditional on his own valuation being one of the Q highest.
- If the expected price from the qs+1 item exceeds that of the qs item before either is auctioned, then we would expect this to cause more (less) aggressive bidding for qs item (qs+1 item) to get a better deal, thus driving up (down) its price.

Multiunit Dutch auction

- To conduct a Dutch auction the auctioneer successively posts limit orders, reducing the limit order price of the good until all the units have been bought by bidders making market orders.
- Note that in a descending auction, objects for sale might not be identical. The bidder willing to pay the highest price chooses the object he ranks most highly, and the price continues to fall until all the objects are sold.

Clusters of trades

- As the price falls in a Dutch auction for Q units, no one adjusts her reservation bid, until it reaches the highest bid.
- At that point the chance of winning one of the remaining units falls. Players left in the auction reduce the amount of surplus they would obtain in the event of a win, and increase their reservation bids.
- Consequently the remaining successful bids are clustered (and trading is brisk) relative to the empirical probability distribution of the valuations themselves.
- Hence the Nash equilibrium solution to this auction creates the impression of a frenzied grab for the asset, as herd like instincts prevail.

Why does the Dutch auction not satisfy revenue equivalence?

- We found that the revenue equivalence theorem applies to multiunit auctions if each bidder only wants one item, providing the mechanism ensures the items are sold to the bidders who have the highest valuations.
- In contrast to a single unit auction, the multiunit Dutch auction does not meet the conditions for revenue equivalence, because of the possibility of “rational herding”.
- If there is herding we cannot guarantee the highest valuation bidders will be auction winners.

Asymmetric valuations

- In a private valuation auctions the bidders have different uses for the auctioned object, and this fact is common knowledge to every bidder.
- Each bidder knows the that everyone else is drawing their valuations from the same probability distribution, and uses that information when making her bid.
- What happens if the private valuations of bidders are not drawn from the same probability distribution function?
- In that case the revenue equivalence theorem is not valid, and the auctioneer's prefers some types of auctions over others.

Bidding with differential information

- For example one bidder might know more about the value of the object being auctioned than the others.
- What happens if they are asymmetrically informed about a common value?
- An extreme form of dependent signals occurs when one bidder know the signal and the others do not. How should an informed player bid? What about an uninformed player?

Second price sealed bid auctions

- In a second price sealed bid auction, Rule 3 implies the informed player optimally bids his true value.
- The uninformed player bids any pure or mixed distribution. If he wins the auction he pays the common value, if he loses he pays nothing, and therefore makes neither gains or losses on any bid.
- This implies the revenue from the auction is indeterminate.

Perspective of the less informed bidder in a first price auction

- Suppose the uninformed bidder always makes the same positive bid, denoted bfixed. This is an example of a pure strategy.
- Is this pure strategy part of a Nash equilibrium?
- The best response of the informed bidder is to bid a little more than bfixed when the value of the object v is worth more than bfixed, and less than bfixed otherwise.
- Therefore the uninformed bidder makes an expected loss by playing a pure strategy in this auction. A better strategy would be to bid nothing.

Equilibrium bidding

- The argument in the previous slide shows that the uninformed bidder plays a mixed strategy in this game.
- One can show that in equilibrium when the auctioned item is worth v the informed bidder bids:

(v) = E[V|V v]

- Furthermore the uninformed bidder chooses a bid at random from the interval [0, E[V]] according to the probability distribution H defined by:

H(b) = Prob[(v) b]

Return to the uninformed bidder

- If the uninformed player bids more than E[V], then his expected return is negative, since he would win the auction every time v < E[V] but less frequently when v > E[V].
- We now show that if his bid b < E[V], his expected return is zero, and therefore any bid b < E[V] is a best response to the informed player’s bid.
- If the uniformed bids less than E[V] and loses the auction, his return is zero. If he bids less than E[V], and wins the auction, his return is:

E[V| (V) < b] – b = E[V| V < -1(b) ] – b

= (-1(b) ) – b

= 0

Return to the informed bidder

- Since the uninformed player bids less than E[v] with unit probability, so does the informed player.
- Noting that (w) varies from v to E[v], we prove it is better to bid (v) rather than (w). Given a valuation of v, the expected net benefit from bidding (w) is:

H((w))[v - (w)] = Pr{V w}[v - (w)] = P(w)[v - (w)]

- Differentiating with respect to w, using derivations found on the next slide, yields P’(w)[v - w] which is positive for all v > w and negative for all v < w, and zero at v = w. Therefore bidding (v) is optimal for the informed bidder with valuation v.

The derivative

- Noting:

it follows from the fundamental theorem of calculus that:

and so the derivative of P(w)[v - (w)] with respect to w is:

When does revenue equivalence fail?

- The valuations of bidders might be drawn from different probability distributions, as in the previous example.
- The theorem does not apply when values are not private, as in the previous example.
- Nor does it apply when bidders are interested in buying more than one unit each.
- Moreover block traders attempt to hide their large demands hoping the market maker will not adjust the spread unfavorably. The greater the number of units demanded, the higher the competitive bids, and hence the reason for bidders hiding their demands.

Lecture Summary

- Auctions are amongst the simplest of trading mechanisms.
- We analyzed strategic and revenue equivalence.
- We proved what to bid when revenue equivalence holds.
- Our analysis suggests two guiding principles for bidding:
- Assume your own valuation is high enough to win, and bid high enough to pay the break-even value of the highest losing bid.
- Shave your bid to account for the winner’s curse if you don’t know your own value.
- Our proof of the optimal bidding rule for auctions with differential information illustrates how to proceed on a case by case basis when revenue equivalence fails.

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