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Games, Information and Strategy(14/16)

2012/05/31

NTU MBA

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Integrative Framework

Non-cooperative Games

Cooperative Games &Negotiation

Mechanism Design &Deal Structuring

Strategic Moves:(Compellence, Deterrence)

X (Promises, Threats)

Socially Optimal?

Information Asymmetry

Institution

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Course Schedule

Parts 1 & 2 Sessions 1-5

Part 3

Sessions 6-9 Midterm Exam (2 hours)

Session 10 (04/26, 7-9pm)

Covering Ch1-10, slides, movie Dr. Strangelove(04/19)

Parts 3 & 4 and Other Applications Sessions 11-16

Final Exam Session 17 or 18

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Main Themes (1/2)

Part 1: Introduction and general principles

Basic ideas and examples

How to think about strategic games

Part 2: Concepts and techniques

Rationality and equilibrium

Non-cooperative and cooperative games

Sequential and simultaneous moves Discrete and continuous strategies

Zero-sum and non-zero-sum games

Pure and mixed strategies

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Main Themes (2/2)

Part 3: Some broad classes of games and strategies Uncertainty and information

Strategic moves

Prisoners dilemma and repeated games

Collective-action games

Evolutionary games

Mechanism design

Part 4: Applications to specific strategic situations

Brinkmanship Strategy and voting

Bidding strategy and auction design

Bargaining

Markets and competition

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Strategy and Voting

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Social Choice

Individual preferences are aggregated to produce asocial welfare function - essentially a preferenceranking of the scenarios that are possible to society.Social choice theory is the philosophical and

mathematical study of the type of conclusions that candetermined through various aggregation methods. Using different metric such as interests, values and

welfare, social choice theory aims to determine theoptimal rules of structuring a fair votingframework. Social choice theory is a growing disciplinestarted by Keneth Arrow who originated the studyfollowing his introduction of the impossibility theoremin 1951. This theory is applicable to group decisionmaking, negotiationsand other economic processes.

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Democracy

Participation Voting: necessary condition Purpose: to avoid suppression

Freedom

Natural rights? Premise for participation Political, religious, economic, speech Self-defense Purpose: efficiency and innovation

Equity Premise for voting Law? Procedure? Opportunity? Property? Moral appeal Against inequity

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Ends and Means

Ends (objectives)

Justice, ideal

Self-fulfillment, self-dignity, self-control

Means (methods)

Voting and Social choice

Examples

Plato Philosopher King: Education? Institution?

Karl Marx Dictatorship of the proletariat()

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Democratic Institution

Republic (Madison)

(popularness): (for participation andequity) :(for freedom)

No other conditions! We deserve it! Liberal () and populist ()

interpretation of voting

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Main Themes

Voting rules and procedures and somenonintuitive or paradoxical results that canarise in certain situations

Criteria used to judge the various votingmethods

Strategic behavior of voters and the scope for

outcome manipulation Median voter theorem

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Voting Rules and Procedures

Binary Methods Majority rule Pairwise voting and multistage Condorcet method: a complete round-robin of majority rule Copeland index: first round of the World Cup soccer tournament Amendment procedure (pairwise procedure for three possible alternatives)

Plurative methods Positional methods: assigning points Plurality rule: single vote for most-preferred alternative Antiplurality method Borda count Approval voting

Mixed methods

Majority runoff (2 stages) Successive rounds to eliminate the worst alternative until two alternatives

remain for final majority runoff Single transferable vote Proportional representation Hare procedure (single transferable vote)

Strategic voting (strategic misrepresentation of preferences)

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Voting Paradoxes

The Condorcet Paradox Intransitive ordering

The agenda paradox Example: binary voting procedure for three alternatives Setting the agenda is the real game (e.g., the appointment or

election of the chair) The reversal paradox

In an election with at least 4 alternatives when one is removed

Change the voting method, change the outcome 100 voters for candidates A, B and C

Preference order: 40 voters (A > B > C), 25 voters (B > C >A), 35 voters (C > B > A) Simple plurality rule: winner A, (A:B:C = 40:25:35) Borda count (3-2-1 points): winner B, (A:B:C =

180:225:195) Majority runoff: winner C, (A, B, C) (A, C) C

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Individual ranking(4 = best)

Totalpoints

X Y Z

A 4 4 1 9

C 3 3 4 10

D 2 1 3 6

E 1 2 2 5

Individual ranking

(5 = best)Totalpoints

X Y Z

A 5 5 2 12

B 4 4 1 9

C 3 3 5 11

D 2 1 4 7

E 1 2 3 6

Preferences over Welfare Policies

Left Center Right

Generous Average Limited

Average Limited Generous

Limited Generous Average

(Intransitive ordering)

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Vote- or Preference Aggregation Method

Kenneth Arrows Impossibility Theorem No preference aggregation method satisfies all six of the critical principles at

the same time:1. The social or group ranking must rank all alternatives (be complete).2. It must be transitive.3. It should satisfy a condition known as positive responsiveness, or the Pareto

property in which unanimous preferences within the electorate are reflected in

the aggregate ranking.4. The ranking must not be imposed by external considerations (such as customers)

independent of the preferences of individuals in the society.5. It must not be dictatorial no single voter should determine the group ranking.6. And it should be independent of irrelevant alternatives that is, no change in the

set of candidates (addition to or subtraction from) should change the rankings ofthe unaffected candidates.

The theorem is abbreviated by imposing the first four conditions and focusingon the difficulty of simultaneously obtaining the last two; the simplified formstates that we cannot have independence of irrelevant alternatives (IIA)without dictatorship.

No truly fair voting method can be found without violating some apparentlyreasonable aggregation requirement.

IIA: violated by single transferable-vote procedure, Borda count Relax Arrows principles? Compromise?

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Concerning Paradoxes

Blacks condition (by Duncan Black) Single-peaked preferences: restriction to prevent the Condorcet paradox Condition: alternatives can be ordered along some specific dimension Pairwise (majority) voting satisfies Arrows transitivity condition

Robustness (by Partha Dasgupta and Eric Maskin) Measured by considering how often a voting procedure that is nondictatorial and that satisfies

IIA as well as the Pareto property also satisfies the requirement of transitivity of its social

ranking Simple majority rule: maximally robust Borda count Plurality rule

Intensity ranking (Donald Saari)6 IBI (intensity of binary independence): Societys relative ranking of any two alternatives should

be determined only by (1) each voters relative ranking of the pair and (2) the intensity of thisranking

Weaker than IIA: applying IIA only to such additions or deletions of irrelevant alternatives thatdo not change the intensity of peoples preferences between the relevant ones Borda count: ties can occur through Condorset terms prevented by single-peak preferences) or

reversal terms (prevented by using only ballots for the full set of candidates)

Condorcet criterion Consistency criterion Lack of manipulability

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Strategic Manipulation of Voting

Plurality rule Spoiler: Ross Perot in 1992 U.S. presidential election (Bush-Clinton-Perot);

Ralph Nader in 2000 election (Gore-Bush-Nader) Plurality in separate constituencies vs. proportional representation of the

whole population: ( two major parties and tyranny of the majority) vs. (several parties and inconclusive bargaining of power and legislative gridlock)

Britain: Conservative-Labor-Liberal; Italy: often no party has a clear majorityof seats

Pairwise voting Agenda-setting chair Voters misrepresentation of preferences

Borda Count

Remedy: Ballots are open to public scrutiny in sports award elections (usingthe fear of public reprisal to keep voters honest in the elections).

With incomplete information Strategy conditioned on the beliefs about the distribution of the different

preference orderings among the other voters and on the beliefs about howtruthful other voters will be

Truthful voting is a dominant strategy for all voters in the final round.

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For A:

Center

A G

LeftA A A

G A G

For G:

Center

A G

LeftA A G

G G G

For L: CenterA G

LeftA A A

G A G

For G: CenterA G

LeftA A A

G A G

For A;Center

A L

LeftA G G

L G L

For L:Center

A L

LeftA G L

L L L

Election Outcomes in Two Possible Second-Round Votes

Election Outcomes Based on First-Round Votes

Right votes

Right votes

Right votes

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Concerning Manipulation

Arrows theorem Does not require nonmanipulability

William Vickrey Procedures satisfying Arrows IIA assumption most immune to strategic manipulation The situations with small numbers of informed voters and smaller sets of available alternatives

may be most susceptible to manipulation, given a voting method that is itself manipulable. Weakening the IIA assumption to help voting procedures satisfy Arrows conditions makes way

for more manipulable procedures. Gibbard-Satterthwaite theorem

If there are three or more alternatives to consider, the only voting procedure that preventsstrategic voting is dictatorship: one vote is assigned the role of dictator and his preferencesdetermine the election outcome.

Arrows theorem Often reduced to a consideration of which procedures can simultaneously satisfy

nondictatoship and IIA

In decreasing order of manipulability Approval voting, the Borda count, the amendment procedure, majority rule, the Hare procedure(single transferable vote)

Depends only on the amount of information necessary to manipulate a voting system and isnot based on the ease of putting such information to good use or whether manipulation is mosteasily achieved by individual voters or groups

In practice, the manipulation of plurality rule by individual voters is quite difficult

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The Median Voter Theorem

In election with only 2 candidates

Principle of minimum differentiation

Discrete political spectrum

Symmetric distribution Asymmetric distribution

Continuous political spectrum Uniform distribution

Normal distribution Limitation

It applies when there is just one issue, or on a one-dimensional spectrum of political differences.

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http://www.amazon.com/gp/reader/0300013647/ref=sib_dp_pt

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Bidding Strategy & Auction Design

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Types of Auctions

Auction rules Open outcry

English or ascending auction (with or without auctioneer): Christies,Sothebys, Internet

Dutch or descending auction (using clock): flowers, fish

Sealed bid First-price auction Second-price auction (Vickrey auction)

Others The highest bidder wins but the top two bidders pay their bids The highest bidder wins but all bidders pay their bids

Auction environments Valuation Common-value or objective-value auction: oil-drilling tract, interest rate

for bond traders Private-value or subjective-value auction: antique a necklace worn by

Princess Diana

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The Winners Curse

In common-value auction

Example: bidding for a company Accepted bidding price b; real worth between 0 and b; average worth b/2

Worth 50% more under your control; 1.5 * b/2 = 0.75b < b still

Why existing management not bid against you?

You will win the takeover battle only when it is not worth winning?

Advice of two questions Would I be willing to purchase the tract of land for $1.08 billion, given

what I know before submitting my bid?

Would I still be willing to purchase the tract of land for $1.08 billion,

given what I know before submitting my bid andgiven the knowledgethat I will be able to purchase the land only if no one else is willing to bid$1.08 billion for it?

Similar to adverse selection How much should we shade down our bidding to take into account the

winners curse?

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Bidding Strategies

Standard open-outcry English auction The valuation of the second-highest bidder (difference:

the minimum bid increment defined in the auction rule)

First-price sealed-bid auction Shading bid (increases profit margin if success but

lowers chances of being the high bidder)

Dutch auction Similar to first-price sealed-bid auction

Second-price sealed-bid auction Truthful bidding (dominant strategy)

Vickreys Truth Serum: truthful bidding

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All-Pay Auctions

Examples Olympic games Political contests Lottery

Rule Common-value, sealed-bid, first-price auction in which every bidder pays his

bid. Your bid is wasted unless you win. The sum of all the bids often exceeds the value of the prize by a largeamount.

Destructive competition? No bidding cannot be an equilibrium

Mixed strategy nbidders Common-value object worth 1 Bid x, a continuous variable in the interval (0, 1) [P(x)]n-1 = x,or P(x) = x1/(n -1) n= 2, equilibrium strategy bid: random and uniformly distributed over the

whole range from 0 to 1 n= 3, the prob of bidding 4 or less is 1/2 Any players average or expected bid: 1/n

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How to Sell at Auction (1/3)

Auction as a sequential-play game Setting of the rule, then bidding

Setting a reserve price

Auction environment is critical to seller revenue Truthful bidding

Bidders attitudes toward risk and their belief about thevalue

Possibility of collusion among bidders

Political considerations when selling public property(drilling rights)

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Risk-neutral bidders and independent estimates Least complex, no winners curse Revenue equivalence: Four auction methods would yield

same expected prices. Advanced mathematical techniques needed to prove that revenue

equivalence can be extended to Dutch and first-price, sealed-bidauctions

Experimental evidence Dutch auction prices lower on average than first-price, sealed-bid

auction prices (some positive utility associated with suspensionfactor?)

Overbidding in second-price, sealed-bid auctions but not in Englishauctions (When people have to specify a price, these sealed-bid

auctions seem to draw more attention to the relationship betweenthe bid price and the probability of ultimately winning the item) Field evidence from Internet auctions

Dutch auction revenue as much as 30% higher on average than first-price, sealed-bid revenue. (additional bidder interest or impatiencein the course of a 5-day auction)

Near revenue equivalence for the other two auction types

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Risk-averse bidders (with bids and beliefsuncorrelated) Bidders generally want to win, more concerned about the

losses caused by underbidding losing the objects than bythe costs associated with bidding at or close to their truevaluation

First-price sealed-bid auction: shading down bids less Dutch auction: waiting less to bid

Correlated estimates Relevant for common-value auctions Positively correlated (assuming a large number of bidders)

Pessimism: second-price, sealed-bid auction (shading down bidsmore in first-price, sealed-bid auction) Optimism: English auction (in the feverish bidding process)

Negatively correlated (small number of bidders with potentiallyvery different valuations) Dutch auction First-price sealed-bid structure

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Some Added Twists to Consider

Multiple objects Independent valuation or package valuation Example: real-estate development

A (square in shape) : 4 sq acres = 16 quarter-acre blocks : corner lots lots on eachside

B: 4 sq acres = a full block 2 individual 2-acre lots 4 1-acre lots Developers bidding strategy? Risk of being beaten in just one round?

Defeating the system Vickrey auction Collusion among bidders (to submit low second-highest bid) Countered by setting reserve prices

First-price sealed-bid less vulnerable to bidder collusion Multi-person prisoners dilemma Cheating more difficult to detect Sustaining collusion in repeated game

Shilling: Seller planting false bids to inflate the final bid price (in English auction

and in second-price, sealed bid auction) Information disclosure on sellers side

Sellers private information (about a common-value object) Concealment even of favorable information usually deemed as a negative signal to buyers Disclosure reduces bidders shading and the effects of winners curse Honesty is often the best policy

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Auctions on the Internet

eBay as No. 1 Listing fees, seller commissions, advertising revenues

Most useful for selling goods available in limited quantity There is unknown demand The seller cannot easily determine an appropriate price Matching process to find market prices

Auction rules English auction and second-price, sealed bid popular Proxy-bidding like second-price sealed-bid (auction site displaying only one

bid increment rather than the reservation price) Dutch auction rare in use. Only a few retail sites (Lands End) posting with

successively reducing prices for overstocked items Yankee auctions (for multiple identical units in a single auction): each paying

bid price per unit or (with Dutch auction label) the lowest winning bid price

Auction-ending rules Most often 7 days Hard end time (eBay) Soft end time: extension if a new bid received within 10 mins (Amazon) Bidders hiding private information and bidding late (sniping). So eBay

preferable Sellers may get better prices at Amazon. Bidders bid earlier at Amazon

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The Airwave Spectrum Auctions (U.S.)

Designing the rules Prevention of monopoly and promotion of small businesses Open auctions: benefit of efficiency and positively correlated estimates vs.

cost from collusion Simultaneous multiple rounds No package (or combinatorial) bidding; maybe in future

Reserve prices Deposit (20% of bid) to bid as down payment (within 5 days of the auction)

and to incur penalties for defaulting on a winning bid Installment payment plans bidding credits of 25% to 35% for small

businesses Entrepreneurs blocks held for bidders under a given financial level

How the auctions have fared Government total revenue $23 billion (1994-1998), $78 billion (as of Nov.

2008)

Two specific bidding issues Default: increased competition and overbidding Bid rigging: code bidding (last 3 digits) and electronic system

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05/31 Assignment

Ch 16 & 17: Solved questions

http://www.wwnorton.com/studyspace/disciplines/economics.aspx?DiscId=6

Ch 16 & 17: Unsolved questions (optional)
http://www.wwnorton.com/studyspace/disciplines/economics.aspx?DiscId=6http://www.wwnorton.com/studyspace/disciplines/economics.aspx?DiscId=6http://www.wwnorton.com/studyspace/disciplines/economics.aspx?DiscId=6http://www.wwnorton.com/studyspace/disciplines/economics.aspx?DiscId=6

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