URBS 4/511 Urban Policy & Strategic Analysis

The Art of Strategy

Avinash K. Dixit & Barry J. Nalebuff

NY:  W.W. Norton, 2008


Part One

Ch. 2  Games Solvable by Backward Reasoning

  1. “Game”—situation of strategic interdependence
    1. May be conflict (zero-sum; one’s gain is the other’s loss); more commonly, games are combination of mutually gainful (or harmful) strategies
    2. May be sequential or simultaneous
    3. Different games may have identical or very similar mathematical forms
    4. Port of the interdependence is the credibility of the players
    5. It is not always the case that more for one layer means less for others
  2. Sequential games: 
    1. use decision trees  to analyze.
    2. Rule 1:  Look forward & reason backward
    3. “Advantage of commitment:--limiting one’s own freedom of action in a sequential game to influence the other player’s actions
    4. To be “fully solvable” by backward reasoning, a sequential game must have

                                                               i.      Absence of uncertainty

                                                             ii.      Situation is completely determined (no chance determinants)

                                                            iii.      Other’s objectives are known (but do not assume they are the same as yours)

    1. Problem of “strategic uncertainty” (How will the other player react?)

                                                               i.      Need to balance “self-regarding” & “other-regarding” behavior


Ch. 3  Prisoners’ Dilemmas and How to Resolve Them

  1. Simultaneous games:  Use a payoff matrix to calculate strategy.
    1. Payoff matrix—lay out all the consequences of all the combinations of the simultaneous choices. 
    2. (See Gambit software, http://gambit.sourceforge.net and ComLabGames, http://www.comlabgames.com) 
    3. Dominant strategy—a strategy that is better for a player than all other available strategies, no matter what strategy or strategy combinations the other player chooses.
    4. “Quasi-magical thinking” (Shafir & Tversky)—idea that by taking some action you can influence what the other will do, even if decisions are being made simultaneously.
  2. Rule 2:  If you have a dominant strategy, use it.
  3. Conventional economic theory from Adam Smith on is wrong to assume that pursuing one’s own self-interest leads to an outcome that is best for all.
    1. “General theory of tacit cooperation in repeated games”—Cheating may gain one layer a short-term advantage, but this harms the relationship and creates long-run costs.
  4. One Strategy for playing a Prisoners’ Dilemma
    1. Principles for effective strategy

                                                               i.      Clarity

                                                             ii.      Niceness

                                                            iii.      Provocability

                                                           iv.      Forgivingness

    1. Tit for tat:  cooperate in first period and from then on mimic rival’s action from previous period.           

                                                               i.      At best, ties its rival

                                                             ii.      Always comes close to rival strategies

                                                            iii.      At worst, ends up getting beaten by one defection

                                                           iv.      “Flawed” strategy

1.      mistake “echoes” back & forth

2.      will never accept punishment without hitting back—creates “cycles” or “reprisals”

3.      no way to say “enough is enough”

  1. But cooperation occurs significantly often, even with one-time games.
    1. Women cooperate with each other more, less with men, and men least of all.
    2. “Contribution game”—initial stake, one can keep part, rest goes to common pool.  Pool is then doubled and divided equally.

                                                               i.      People will punish “social cheaters”

                                                             ii.      prospect of punishment increases contributions in the first place

    1. How to achieve cooperation

                                                               i.      detection of cheating (as immediate—and targeted—as possible)

                                                             ii.      nature of punishment (external or intrinsic)

                                                            iii.      clarity (boundaries and consequences should be clear from the outset)

                                                           iv.      certainty (of both reward and punishment)

                                                             v.      size (“principle of graduation”—let the punishment fit the crime)

                                                           vi.      repetition (discounting of future benefits, likelihood of continuing relationship)

    1. Dilemma of collusion

                                                               i.      Contributes to clarity among the players

                                                             ii.      Unnecessary punishment among the players is avoided

                                                            iii.      But it harms the general public’s interest


Chapter 4  A Beautiful Equilibrium

  1. Nash Equilibrium—a game outcome where the action of each player is best for him given his beliefs about the other’s action, and the action of each is consistent with the other’s beliefs about it.
    1. “Prominence”—where there are multiple Nash equilibria, one of the strategies must become more salient than the others.
    2. “focal point”—when players’ expectations converge on a prominence
    3. What is important is not that it is obvious to you, or to your partner, but that it is obvious to each that it is obvious to the others.
    4. As a result, equilibrium can easily be determined by whim or fad
  2. Studying Cases Using  Nash Equilibira
    1. Simplify the analysis

                                                               i.      Rule 3:  Eliminate dominated strategies from consideration.

                                                             ii.      Remove all dominated strategies and all “never best” strategies

                                                            iii.      Look for “mutual best” response cells

                                                           iv.      Rule 4:  Look for an equilibrium, a pair of strategies in which each player’s action is the best response to the other’s.

                                                             v.      If there is a unique equilibrium of this kind, there are good arguments why all players should choose it.

    1. Can be solved mathematically (in some cases) by solving multiple linear equations for their point(s) of intersection (“linear programming”).
    2. Begin with Nash equilibrium, then think of reasons why, and manner in which, outcomes may differ from Nash predictions.

                                                               i.      Requires not only choosing best response based on belief about the other players’ actions, but also requires that those beliefs are correct!


Part Two

Ch. 5  Choice and Chance

  1. If you follow any system or pattern in your choice of strategy, it will be exploited by the other player to his advantage and to your disadvantage.
    1. The value of randomization can be quantified (soccer free-kick example)
    2. In a strict zero-sum game, one should attempt to

                                                               i.      Minimize the opponent’s maximum payoff (minimax), while

                                                             ii.      The opponent attempts to maximize his own minimum payoff (maximin)

                                                            iii.      If both do this, minimax equals maximin (vonNeumann-Morgenstern theorem—which is a special case of the Nash equilibrium)

    1. Rule 5:  In games of pure conflict, choose at random from your available pure strategies.

                                                               i.      Cannot rely on opponent to randomize—you need to use your best mix to keep your opponent using his.

                                                             ii.      In one-shot games, keep options open as long as possible, and choose at the last possible moment.

  1. Limits to “Mixed Strategies”
    1. Many real-world games are not zero-sum
    2. Difficult to accept leaving outcomes to chance when the corporate culture is based on maintaining control
    3. Most commonly used to motivate compliance at a lower monitoring cost


Ch. 6  Strategic Moves

  1. “Strategic moves” are actions that change the game to ensure a better outcome for the player taking the action.
    1. For a response rule to work, it must

                                                               i.      seize the first-mover status to declare a planned course of action

                                                             ii.      be credible (when the time comes, you will actually choose it—which implies that it will be your best choice in that situation)

    1. What needs to be done

                                                               i.      “Commitment”—unconditional strategic move that restricts one’s future actions

                                                             ii.      “Threats & promises”—conditional strategic moves that fix in advance a response rule, even when they require one to act against one’s own interest.

1.      “Threat”—response rule that punishes players who fail to act as desired

2.      “Promise”—response rule that offers to reward others who act as desired

                                                            iii.      Deterrence  (stop others from doing something that otherwise they would do) or Compellence (cause others to do something they otherwise would not do).

                                                           iv.      Response rules that do not promise a change in behavior may be

1.      Warning—showing a self-interest to carry out a threat

2.      Assurance—showing a self-interest to carry out a promise

    1. Other Players’ Strategic Moves

                                                               i.      In sequential games, if there is a second-mover advantage, one can benefit by arranging that the opponent moves first.

                                                             ii.      Other times, goal will be to prevent opponent from making an unconditional commitment (Sun Tzu’s advice)

                                                            iii.      It is never advantageous to allow others to threaten you

    1. Similarities and Differences Between Threats and Promises

                                                               i.      Distinction between threat & promise depends on what is called the status quo

1.      a compellent threat is like a deterrent promise, with a change of status quo.

2.      Cost:  A threat can be less costly (especially if it is successful); a promise, if successful, must be fulfilled.

3.      Purpose: 

a.       Deterrence has no timeline, and can be achieved more simply and better than a threat. 

b.      Compellence has a deadline (and can be defeated by “salami tactics”).  One’s purpose is better achieved by a promise.

    1. Clarity and Certainty

                                                               i.      Clarity--To make a threat or promise, one must communicate clearly to the other player what actions will bring what punishment (or reward)

                                                             ii.      Certainty--The other player must believe the threat or promise

1.      This need not deny the presence of risk

2.      May be implemented in multiple small steps (to avoid salami tactics).

3.      Keep threats at the smallest level needed to keep them effective (“principle of graduation,” remember?)

4.      Brinksmanship—start small, but deliberately and gradually raise the size of the threat (risk)

a.       Increase the risk of the (same) bad thing happening

b.      Most threats carry a risk of error, and therefore an element of brinksmanship

c.       Even with the best of care, brinksmanship may fail (the bad thing may actually happen)

    1. How to do it (making it credible)—see Ch. 7


Ch. 7  Making Strategies Credible

  1. Words must be  backed with appropriate strategic actions if they are to have an effect on the other players’ beliefs and actions (unless their objectives are perfectly aligned with yours, in which case you can trust their words).
    1. Change the payoffs of the game (make it in your interest to follow through on your commitment—turn a threat into a warning, a promise into an assurance)

                                                               i.      Write contracts to back up your resolve

1.      enforcing party must have some independent incentive to do so

                                                             ii.      Establish and use reputation

    1. Change the game by limiting your ability to back out of a commitment

                                                               i.      Cut off communication (can make action truly irreversible)

                                                             ii.      Burn bridges behind you

                                                            iii.      Leave the outcome beyond your control (or even to chance)

                                                           iv.      Move in small steps

1.      lessen the risk from breaking the contract

2.      use when large degree of commitment is infeasible

3.      to avoid unraveling of trust, there should be no clear final step

    1. Use others to help you maintain your commitment

                                                               i.      Develop credibility through teamwork

                                                             ii.      Employ mandated agents

1.      especially useful if other bonds of friendship or social links exist that you are reluctant to break

  1. Undermining Other’s Credibility
    1. If the other player’s strategic moves can hurt you, it is in one’s self interest to prevent those moves from being credible.  The same techniques apply:

                                                               i.      Contract

                                                             ii.      Reputation (you can neutralize the other’s reputation by keeping it secret)

                                                            iii.      Communication (one can be unavailable to receive a communication)

                                                           iv.      Burning bridges

                                                             v.      Moving in steps (resist in small steps—salami tactics)

                                                           vi.      Mandated agents (refuse to deal with the agent and demand to speak to the principal)


Part Three


Ch. 8  Interpreting and Manipulating Information

  1. Why not rely on others to tell the truth?  The answer is obvious: because it might be against their interests.
    1. The greater the conflict, the less the message can be trusted
    2. “Second level deception”—lying by telling the truth
  2. Actions speak louder than words.  Watch what the other play does, not what he or she says.
    1. So, each player will try to manipulate actions for their information content.
    2. “Signaling”--Choosing actions that promote favorable leakage
    3. “Signal jamming”—Choosing actions that reduce or eliminate unfavorable leakages
    4. “Screening”—Setting up a situation where the person would find it optimal to take one action if the information was of one kind, and another action if it was of another kind.
    5. To be an effective signal, an action should be incapable of being mimicked by a rational liar

                                                               i.      It must be unprofitable when the truth differs from what you want to convey

  1. “Information asymmetry”
    1. Screen and/or signal by letting people make choices from a suitably designed menu

                                                               i.      Works because the cost of using the device is less for those you want to attract than for those you want to avoid

                                                             ii.      Action serves to discriminate between possible types

    1. People have better information about their own risks than do their potential partners (e.g., health insurance)
    2. “Adverse selection”—Groucho Marx effect.  “I wouldn’t want to belong to any club that would have me.”
    3. “Positive selection”—Capital One strategy.  No one but the type of person you want would be interested.
    4. Examples in use:

                                                               i.      Bureaucratic delay may not be due to inefficiency, but a strategy for coping with information asymmetry.

                                                             ii.      Benefits in kind can serve as a screening device

                                                            iii.      “The dog that doesn’t bark”—not sending a signal also sends a signal.  (E.g., an opportunity to invest could imply a requirement to coinvest)

                                                           iv.      Countersignaling—sometimes the most powerful signal you send is that you don’t need to signal (nouveau riche versus old money)

    1. Signal jamming

                                                               i.      Pooling equilibrium (of signaling game)—all types take the same action, rendering the signal uninformative

                                                             ii.      Separating equilibrium—one type signals and others do not, thus distinguishing the types.

                                                            iii.      Most actions convey only partial information (they are “semi-separating”).  As a result, there is a probabilistic element to the information obtained.

1.      You can know the likelihood of a distinction, but not the true character of a single case (probabilities, not payoffs)

2.      Bayes’ Rule is used to calculate probabilities.

  1. Price Discrimination by Screening
    1. Create different versions of the same good and price the versions differently.
    2. “Incentive compatibility constraint”—Cannot charge full willingness to pay for higher-priced good (since there is a benefit to defecting to lower-priced good)
    3. “Participation constraint”—Cannot raise lowest price higher than that type’s willingness to pay


Ch. 9  Cooperation and Coordination

  1. Uncoordinated choices can interact to produce a poor outcome for society.  Even in a situation of relative (rather than absolute) performance, the game does not have to be zero-sum.
    1. “Scope of gains” can be increased by reducing the inputs.
    2. But this would require an enforceable collective agreement
    3. It is difficult to arrange a self-enforcing cartel; it is better if an outsider enforces the collective agreement (such as limiting competition)
    4. One can use the market to charge people for the harm (or negative benefit) they cause to others
    5. There are often three equilibria—one at either pole of the choices, and a mid-range.  The social dynamics that drive toward one of the extremes is called “tipping”

                                                               i.      Guarantees can be offered to offset social dynamics and stabilize a mid-range equilibrium

    1. Piecemeal action can result in less than optimal outcomes because of the voting paradox (pairwise comparison of votes can result in no winner)—it ignores the intensity of preferences.

                                                               i.      Sometimes preferable to consider reforms only as a package rather than a series of small steps

                                                             ii.      Sometimes preferable to fail at a very difficult task—so sometimes it might be preferable to increase your chances of failing in order to reduce the consequences.

  1. Summary:  The Market doesn’t always get it right
    1. Either/or choices

                                                               i.      Everyone makes the same choice, but it is the wrong one

                                                             ii.      Some make one choice, but the proportion is wrong (not optimal) because spillovers on others were not taken into consideration

                                                            iii.      Choice may lead to an extreme equilibrium, rather than a more desirable mid-range

    1. Choice of multiple alternatives

                                                               i.      Choices lead stepwise down a wrong path

                                                             ii.      Excessive homogeneity (mutually reinforcing expectations)

                                                            iii.      No equilibrium is found

    1. Reasons:

                                                               i.      History matters (bandwagon effect)—need a critical mass to change (“tip”) to a more optimal equilibrium

                                                             ii.      Much of what matters is outside the marketplace (unpriced goods have no invisible hand to guide selfish behavior)


Ch. 10  Auctions, Bidding, and Contests

  1. Auction—buyers compete against each other to gain an object, or sellers may compete to get a buyer’s business (“procurement auction”).
    1. English Auction—“ascending auction”

                                                               i.      Bid until the price exceeds your value, then drop out

                                                             ii.      Problem is to determine “value”

1.      private value—independent of others’ assessment

2.      common value—item has same value for all, although each may have a different view as to what that value is.

    1. Japanese Auction—also an “ascending auction”

                                                               i.      All start as bidders, and drop out as value is passed.

                                                             ii.      Winning bidder pays the value at which the pentultimate bidder dropped out.

                                                            iii.      Item is sold to person with highest valuation, seller receives payment equal to second highest valuation.

    1. Vickrey Auction—“second price” auction

                                                               i.      All bids are sealed.  Winning bidder pays price of second highest bid.

                                                             ii.      All bidders have a dominant strategy—bid their true valuation

                                                            iii.      Implications of Vickrey Auctions

1.      Even if you change the rules of the game, players will adapt their strategies and precisely offset those changes.

2.      Online auctions (“proxy bidding”)

3.      “Sniping”—wait until last minute to enter best proxy bid

4.      Attempt to gain strategic advantage by withholding information about one’s true valuation

5.      Sniping may be explained by people not knowing their own valuation.

                                                           iv.      “Winner’s curse”—winning a bid and discovering it isn’t worth what one thought

1.      Solved by “consequentialism”—look ahead for consequences of one’s action, and assume that situation is the relevant one at the time of initial play (“Don’t ask a question if you don’t want to hear the answer.”)

2.      As a result, your bids will often be rejected because you underestimated the value.  But you don’t have to deal with the undervalued good, so it doesn’t matter.

    1. Sealed-Bid Auction

                                                               i.      Seal bid in envelope.  Highest bid wins.

                                                             ii.      Never bid your valuation (or, worse, more than your valuation)

1.      It guarantees that you break even at best

2.      Strategy is dominated by bidding something below your valuation (or, in a procurement auction, bidding a price something above your true price).

    1. Dutch Auction—“reverse auction”

                                                               i.      Auction starts with high price that declines; first bidder to stop the decline pays that price.

                                                             ii.      At optimal bid, savings from paying lower bid is no longer worth increased risk of losing the prize

  1. Implications
    1. “Revenue equivalence theorem”—When valuations are private and game is symmetric, seller makes same amount of money whether auction is English/Japanese, Vickery, Dutch, or sealed-bid
    2. Optimal bidding strategy is to bid what you think the next-highest person’s value is, given the belief that you have the highest value.
    3. As the number of bidders increases, the market approaches perfect competition and all of the surplus goes to the seller
    4. Many games that might not look like auctions turn out to be one:

                                                               i.      Pre-emption game—First person to launch has a chance to own the market, provided they succeed.

1.      wait until you are fully ready and miss the opportunity

2.      jump in too soon and fail

                                                             ii.      War of attrition—instead of who goes in first, game is who gives in first

                                                            iii.      Case study of Spectrum Game

1.      opportunity for tacit cooperation

2.      when two games are combined into one, creates opportunity to employ strategies that go across the two games

3.      can employ punishment/cooperation that would otherwise be impossible without explicit collusion

4.      Moral:  If you don’t like the game, look for the larger game.


Ch. 11  Bargaining

  1. All other things equal, looking ahead and reasoning backward leads to a simple rule:  Split the total down the middle.  But…
    1. Cost of waiting—the better a party can do by itself in the absence of an agreement (it’s BATNA), the larger its share of the bargaining pie will be
    2. Size of the pie (the “total”) is measured by how much value is created when the two sides reach an agreement compared to when they don’t (value above the sum of the BATNA’s)

                                                               i.      Talmudic principle of the divided cloth

                                                             ii.      If  BATNAs are not fixed, opens up strategy of influencing the BATNAs.

    1. Threats/commitments that lower both parties’ outside opportunities can work as long as the rival is damaged more severely (“This will hurt you more than me.”)
    2. Brinksmanship:  May break down, despite both sides’ desire to succeed, because

                                                               i.      Different ideas about what constitutes success

                                                             ii.      Strikes are an example of signaling

                                                            iii.      Brinksmanship (like a strike) is a strategy for the stronger of two parties—the one that fears a breakdown less.

    1. Simultaneous bargaining over many issues

                                                               i.      Put all issues into common bargaining pot, exploit differences in relative valuation to achieve outcomes that are better for everyone.

                                                             ii.      Joining issues together opens possibility of using one bargaining game to generate threats in another

                                                            iii.      “Virtual strike” can work while both sides are still talking.

                                                           iv.      A multiround game allows the receiving side to show that one’s theory of their values is incorrect.

                                                             v.      Rubinstein Bargaining:  stepwise negotiation, taking into account the value of time delay.

1.      Person making proposal has an advantage, proportional to the degree of impatience of the other party.

2.      The lower the cost to waiting, the less the advantage of going first.


Ch. 12  Voting

  1. Voting
    1. Strategic voting paradox:  When your vote doesn’t matter, you should vote with your heart.  But when your vote does matter, you should be strategic.

                                                               i.      It’s okay to speak the truth when it doesn’t matter.

                                                             ii.      Particularly problematic when there are many candidates—polls can become self-fulfilling prophecies (“front-runner effect”)

                                                            iii.      Anyone’s vote affects the outcome only when it creates or breaks a tie.

    1. Naïve voting—in three-way (or more) race, no determinative outcome.
    2. Condorcet rules—each pair of candidates competes in a pairwise vote.  Winner has the smallest maximum votes against.
    3. Voting cycle:  the order in which votes are taken can determine final outcome.
  1. Voters
    1. To distinguish your opinion from the pack, the trick is to take the most extreme stand consistent with still appearing rational
    2. People have an incentive to tell the truth about direction but to exaggerate the intensity of their values.  This is a variant on “split the difference,” where everyone has an incentive to begin with an extreme position for settling disputes.
    3. No voter will take an extreme position if the candidate follows the preferences of the median (not the average) voter

                                                               i.      However, only applicable when choice is one-dimensional.

    1. Majority vote (50% +1) can be easily destabilized; 2/3 majority is stable because the average of all preferences will be 36%.
    2. “Approval voting” (may vote for as many candidates as one wishes) permits voters to express true preferences but still permits a rule (majority of votes cast, limited number of slots to fill) for arriving at a resolution.


Ch. 13  Incentives

  1. Incentive payment schemes must cope with the problem of “moral hazard” (creating a perverse incentive to do something unacceptable)
    1. A fixed-sum does not handle the “incentive” component very well
    2. Pure piece-rate does not handle the participation aspect
    3. System must also be simple enough to be understood.
  2. Writing an incentive contract
    1. Must be based on some observable metric (which is only an imperfect indicator of unobservable effort)

                                                               i.      Always an element of chance in behavior—if chance element is too large, reward will be only poorly related to effort.

                                                             ii.      However, chance is highly correlated across workers.  Incentives can be based on relative performance. (This also works for suppliers)

    1. Quota & bonus versus linear or proportional compensation scheme

                                                               i.      Quota is all-or-none, so does not capture all of effort; also has problem of  setting quota high enough to stretch without being impossible to reach

                                                             ii.      Linear payment is more robust, less prone to manipulation, than proportional

    1. Carrots versus sticks

                                                               i.      Spread of payments (good vs. bad outcomes) determines incentive—wider the spread, more powerful the incentive

                                                             ii.      Average payment must meet the participation constraint (how much could have been earned in other opportunities)

    1. “Efficiency premium”—excess above basic payment from other opportunities
    2. Multiple tasks

                                                               i.      If tasks are substitutes for each other, strong incentive to one will hurt the effort in another. 

                                                             ii.      Therefore, both incentives have to be kept weaker than if each task were considered in isolation.

                                                            iii.      If tasks are complements, a single incentive affect performance in both

    1. Motivation

                                                               i.      Many workers—especially in nonprofits, some public sector, and innovative/creative tasks—are intrinsically motivated when performing tasks that improve their self-image and give them a sense of autonomy.

                                                             ii.      Such tasks need fewer extrinsic motivations (material incentives), and can even diminish the intrinsic motivation.

                                                            iii.      Simple existence of material penalties (lower pay or dismissal for failure) may undermine intrinsic motivation.

                                                           iv.      Should offer significant financial rewards or none at all; small amounts might lead to worst of all outcomes

                                                             v.      Overall strength of incentives is inversely proportional to the number of different supervisors

    1. Data—indicators of quality of effort can be improved by

                                                               i.      Keeping records of individual strings of success/failure

                                                             ii.      Have several experts working on related projects





© 2009 A.J.Filipovitch
Revised 6 June 2009