MdSm 601
Introduction to Modeling & Simulation
Game
Theory
Based on: Avinash K. Dixit
& Barry J. Nalebuff, The Art of Strategy (W.W. Norton, 2008)
Part One
Ch. 2 Games Solvable by Backward Reasoning
- “Game”—situation
of strategic interdependence
- May
be conflict (zero-sum; one’s gain is the other’s loss); more commonly,
games are combination of mutually gainful (or harmful) strategies
- May
be sequential or simultaneous
- Different
games may have identical or very similar mathematical forms
- Part
of the interdependence is the credibility of the players
- It
is not always the case that more for one player means less for others
- Sequential
games:
- use decision trees to analyze.
- Rule 1: Look forward & reason backward
- “Advantage
of commitment:--limiting one’s own freedom of action in a sequential game
to influence the other player’s actions
- 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)
- 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
- The
canonical form of the game:
- Simultaneous
games: Use a payoff matrix to
calculate strategy.
- Payoff
matrix—lay out all the consequences of all the combinations of the
simultaneous choices.
- (See
Gambit software, http://gambit.sourceforge.net
and ComLabGames, http://www.comlabgames.com)
- 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.
- “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.
- Rule 2: If you have a dominant strategy, use it.
- 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.
- “General
theory of tacit cooperation in repeated games”—Cheating may gain one
player a short-term advantage, but this harms the relationship and
creates long-run costs.
- One
Strategy for playing a Prisoners’ Dilemma
- Principles
for effective strategy
i.
Clarity
ii.
Niceness
iii.
Provocability
iv.
Forgivingness
- 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”
- But cooperation occurs significantly
often, even with one-time games.
- Women
cooperate with each other more, less with men, and men least of all.
- “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
- 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)
- 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
- 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.
- “Prominence”—where
there are multiple Nash equilibria, one of the
strategies must become more salient than the others.
- “focal
point”—when players’ expectations converge on a prominence
- 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.
- As a
result, equilibrium can easily be determined by whim or fad
- Studying
Cases Using Nash Equilibira
- 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.
- Can
be solved mathematically (in some cases) by solving multiple linear
equations for their point(s) of intersection (“linear programming”).
- 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
- 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.
- The
value of randomization can be quantified (soccer free-kick example)
- 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)
- 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.
- Limits
to “Mixed Strategies”
- Many
real-world games are not zero-sum
- Difficult
to accept leaving outcomes to chance when the corporate culture is based
on maintaining control
- Most
commonly used to motivate compliance at a lower monitoring cost
Ch.
6 Strategic Moves
- “Strategic
moves” are actions that change the game to ensure a better outcome for the
player taking the action.
- 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)
- 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
- 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
- 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”—taking a small slice at a time until the entire salami is gone). One’s purpose is better achieved by a
promise.
- 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”)
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)
- How
to do it (making it credible)—see Ch. 7
Ch.
7 Making Strategies Credible
- 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).
- 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
- 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
- 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
- Undermining
Other’s Credibility
- 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
- Why
not rely on others to tell the truth?
The answer is obvious: because it might be against their interests.
- The
greater the conflict, the less the message can be trusted
- “Second
level deception”—lying by telling the truth
- Actions
speak louder than words. Watch what
the other play does, not what he or she says.
- So,
each player will try to manipulate actions for their information content.
- “Signaling”--Choosing
actions that promote favorable leakage
- “Signal
jamming”—Choosing actions that reduce or eliminate unfavorable leakages
- “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.
- 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
- “Information
asymmetry”
- 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
- People
have better information about their own risks than do their potential
partners (e.g., health insurance)
- “Adverse
selection”—Groucho Marx effect. “I
wouldn’t want to belong to any club that would have me.”
- “Positive
selection”—Capital One strategy.
No one but the type of person you want would be interested.
- 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)
- 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.
- Price
Discrimination by Screening
- Create
different versions of the same good and price the versions differently.
- “Incentive
compatibility constraint”—Cannot charge full willingness to pay for
higher-priced good (since there is a benefit to defecting to lower-priced
good)
- “Participation
constraint”—Cannot raise lowest price higher than that type’s willingness
to pay
Ch.
9 Cooperation and Coordination
- 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.
- “Scope
of gains” can be increased by reducing the inputs.
- But
this would require an enforceable collective agreement
- It
is difficult to arrange a self-enforcing cartel; it is better if an
outsider enforces the collective agreement (such as limiting competition)
- One
can use the market to charge people for the harm (or negative benefit)
they cause to others
- 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
- 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.
- Summary: The Market doesn’t always get it right
- 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
- Choice
of multiple alternatives
i.
Choices
lead stepwise down a wrong path
ii.
Excessive
homogeneity (mutually reinforcing expectations)
iii.
No
equilibrium is found
- 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
- Auction—buyers
compete against each other to gain an object, or sellers may compete to
get a buyer’s business (“procurement auction”).
- 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.
- 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.
- 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.
- 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).
- 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
- Implications
- “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
- 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.
- As
the number of bidders increases, the market approaches perfect
competition and all of the surplus goes to the seller
- 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
- All
other things equal, looking ahead and reasoning backward leads to a simple
rule: Split the total down the
middle. But…
- Cost
of waiting—the better a party can do by itself in the absence of an
agreement (it’s BATNA—“Best Alternative To Nothing at All”), the larger
its share of the bargaining pie will be
- 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 disputed garment
ii.
If BATNAs are not
fixed, opens up strategy of influencing the BATNAs.
- 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.”)
- 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.
- 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
- Voting
- 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.
- Naïve
voting—but in three-way (or more) race, no determinative outcome.
- Condorcet
rules—each pair of candidates competes in a pairwise vote. Winner has the smallest maximum votes against.
- Voting
cycle: the order in which votes
are taken can determine final outcome.
- Voters
- To
distinguish your opinion from the pack, the trick is to take the most
extreme stand consistent with still appearing rational
- 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.
- 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.
- Majority
vote (50% +1) can be easily destabilized; 2/3 majority is stable because
the average of all preferences will be 36%.
- “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
- Incentive
payment schemes must cope with the problem of “moral hazard” (creating a
perverse incentive to do something unacceptable)
- A
fixed-sum does not handle the “incentive” component very well
- Pure
piece-rate does not handle the participation constraint (how much could
have been earned in other opportunities)
- System
must also be simple enough to be understood.
- Writing
an incentive contract
- 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)
- 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
- 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
- “Efficiency
premium”—excess above basic payment from other opportunities
- 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
- 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
- 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
© 2013 A.J.Filipovitch
Revised 6 January 2013