Presentation by:

Alexander Carter-Silk

THE SCIENCE OF NEGOTIATIONIPTC Group

Game Theory: Its all bunkum !• John Von Neuman Mini Max Strategies• John Nash Nobel Prize for Mathematics• Spence – “ Signalling”• Joseph Stilling Insurance Behaviour”

(screening)• Leonid Hurwicz Nobel Prize for economics• Erik Maskin Nobel Prize• Roger Myeson Nobel Prize• Schelling

“In the British 3g Telecoms Auction, Ken Binmore devised a games strategy which parted the telco’s from $35bn, it was described by Newsweek as the most ruthlessly effective strategy ever deployed in an auction….”

Game Strategies

• The Cortez Strategy• Solomon's Dilemma• Prisoner’s dilema• Battle of the Sexes• Chicken Zero Sum game• Rousseau the Hunter’s Paradox• The Tragedy of the Commons

Ordinal Values & Revealed Preferences

• Does transparency always produce the optimal outcome?

• Threats and Brinkmanship• Contracts• Reputation• Cutting off communication• Moving in steps• Screening• Team work and alliances• Negotiating Agents

TERMINOLOGY

THE GAME:

OPTIMAL PAYOFF:

RATIONAL SELF INTEREST V. ALTRUISMS:

IMPERFECT KNOWLEDGE PERCEPTION v. KNOWLEDGE

CO-OPERATIVE v. NON-CO-OPERATIVE:

UTILITY PREFERENCE AND INDIFFERENCE:

ORDINAL VALUES & THE CHAIN OF INEQUALITIES

THE NASH EQUILIBRIUM :

DOMINANT STRATEGIES NON-DOMINANT STRATEGIES

UNCERTAIN OR IMPROBABLE:

CO-OPERATE OR DEFECT

• There are two players, Row and Column.

• Each has two possible moves,

• “cooperate” or “defect,” corresponding, respectively, to the options of remaining silent or confessing.

• For each possible pair of moves, the payoffs to Row and Column

• R is the “reward” payoff that each player receives if both cooperate.

• P is the “punishment” that each receives if both defect.

• T is the “temptation” that each receives if he alone defects and

• S is the “sucker” payoff that he receives if he alone cooperates.

Chain of Inequalities

Symmetric 2×2 PD With Ordinal PayoffsIn its simplest form the PD is a game described by the payoff matrix:

C D

C R, R S, T

D T, S P, P

satisfying the chain of inequalities:

PD1) T>R>P>S

Temptation, Reward, Punishment, Sucker

D/C = 4/1 C= the sucker bet

DD = 1/1

DD =The wasteland

CC = 2/2

CC = Non-optimal for both parties but better than DD

Game 1: Co-operate or Dominate?

• There is a prize of £1,500.00 to share between the three teams within one minute all of the teams must reach agreement

• Three participants have one minute to reach their optimal conclusion

• The participants must not communicate their desired or optimal pay-off

• Each individual should work towards his/her optimal payoff.

• If no agreement is reached, the participants share the payout, if no agreement is reached each participants get nil (The worst outcome)

The Hunter’s Paradox

Stag Bison Rabbit

Stag 3

3

0

0

1

0

Bison 0

0

3

3

1 0

Rabbit 0

1

0

1

1

1

Dominant Strategy

• The worst outcome (non optimal) is that there is no agreement.

• If my choice is less than the aggregated desired payoffs of the other two participants then I can achieve my pay off by making an open bid

• Assuming uneven payoff aspirations by other participants each greater than 500 any bid above 0 is a strategy achieves better than the worst payoff.

• With an opening bid of anything above 500 the opposing parties must assume that the bid is irrational.

• Multiple parties create co-operation amongst groups (contracts) to improve the probability of an optimal outcome.

• Multiple games which limits the scope for aggression/ defection by the minority.

Game 2 Strategies for Co-Operation

• Each of the participant must choose a number between 1 and 100,

• The number chosen must the same as that chosen by the other participants.

Each participant must consider the choice which each of the others will choose.

That choice must be one which each of the other participants can rationally consider to be the most likely to be chosen by a group which are determined to co-operate.

Game 3: Best Response

• Each participant is to pick a number between 0 and 100 there is a £100 prize for the player whose number is closest to half the other person’s number.

• Some possible dominant strategies– 50– 25– 12.5– 0

• What number will you choose and why?

Predictable behaviour Screening

• There are three types of credit card borrower Max Payers Revolvers and Deadbeats

• The task was to devise a strategy (dominant) which screened out max payers and dead beats

• The answer was the transfer balance option why?– The Revolvers don’t pay interests– The Deadbeats have no intention of paying back– The max payers benefit from zero interest on

balances and rare the most profitable • The strategy has screened out the undesirable card

holders.

Rational v Altruistic

• Two parties begin with equal positions (Ordinal symmetry) an altruistic strategy will provide a better pay off for than a strategy of RATIONAL SELF INTEREST

• One party begins with and Altruistic Strategy and the other with a RATIONAL SELF INTEREST, the party employing the RATIONAL SELF INTEREST will usually dominate.

RATIONAL V. IRRATIONAL

• When one party is motivated by a desire to defeat the opponent rather than minimise risk or optimise payoff

• The probability of success is increased (the

willingness to take the risk of losing).

• The IRRATIONAL dominates the RATIONAL

Probability and Lottery Litigation!

• For any given “state”, there is exists a finite number of outcomes.

• Some of those outcomes are “unlikely”• For any given “utility” for each player there is a

probability that one or more strategies will dominate.

• For any number of players with given combinations of utility and a given number of games there are a limited number of outcomes which are possible

The Bridge: Deterrent or Provocation

• Two cars approach a narrow bridge from opposite directions only one may pass at a time. This requires one car to wait. If both cars enter the bridge simultaneously neither may pass.

• If the cars alternate then both pass but one must wait.

• If both drivers value time (t) equally then either both will lose time or one must act “altruistically” and let the other pass. Both improve, but the one who has acted altruistically gets a less than optimal outcome.

”

Compromise or Co-Operate

• Compromise is not an option it serves neither party, aggression might succeed but now it risks being matched with aggression in which case neither party optimises his out come.

• Both parties do better if one acts altruistically aggression is rewarded, one party must give up his “rational self interest.

• With equal ordinal values and a zero sum game one party must give way to achieve a Nash equilibrium.

• There are two equilibriums A gives way to B and B gives way to A the alternative is the worst outcome for both.

Perception: Rational v Irrational

• The Defender’s perceives that he has a defensible but not impenetrable position

• The Aggressor has an inferior position or force. • The Defendant’s perception of the Aggressor's

tenacity and commitment is material in his (the Defender’s) decision to defend or concede

Cortez

• The Spanish Conqueror Cortez landed in Mexico with a small group of soldiers. To demonstrate that he was committed to victory or nothing, he burnt his ships (no retreat)the Aztecs walked away from the fight.

• Knowledge and perception are not necessarily the same

• The Incas rationalised that if Cortez were willing to fight to the last man, then the best outcome (the dominant strategy) would be to surrender rather than face much greater losses inflicted by an implacable enemy

Threats and Promises

• The best strategic response to a threat of a worst outcome will depend not upon any assessment as to whether the worst outcome=most likely outcome but whether it is perceived that the threat will be carried out.

• Is the outcome a zero sum game.• If it is not a zero sum game is there more

than one possible outcome which represents an equilibrium out come.

Threats and Promises

• Clarity Certainty– Distinguish between compelling and

punishing– Object of a threat is to change behaviour– Object of promise is to reward an optimal

outcome– Is the threat credible.– Earlier performance is rewarded – Threat Late performance is punished.

• Risk proportionate to the threat, increasing the risk. Change the threat or change the rules of the game

Multiple Games – Infinite Games

• If the first game is repeated, several times the dynamics change. The altruistic party will learn that he always loses and may therefore adopt a learnt behaviour strategy.

• Thus

– Move 1 = P1 Cooperates P2 Defects (and crosses the bridge)

– Move 2 = P1 Defects P2 also Defects and his outcome is less than it was in move 1.

– Move 3 = P1 Defects again P1 Co-operates he passes second but does better than he did in move 2

– Move 4 = Two possible nodes P1 Defects (this leads to P2 defecting or P1 co-operates and P2 learns that alternate co-operation defection strategies are not ideal but give the best outcome

Multiple Finite Games

• The game proceeds much as our previous example except the probability of co-operation now changes. Assume a finite number of games N such that for (N-1) the probability of co-operation decreases.

• In this game the party who wishes to optimise his position will consider defecting as the number of games counts down. This party knows;

– He cannot rely on co-operation from the other party as the temptation of increasing the optimal outcome exceeds the probable benefit of co-operation.

– Each party has increased his “utility” from co-operation throughout the “game” and therefore his propensity for risk is increased towards the end of the games. The highest probability on the last move is that both parties will defect

– The increased probability of defection towards the end of the game decreased the effect of “learnt” behaviour.

Imperfect Knowledge• If both parties have different states of knowledge then their

propensity to defect or co-operate are different and the probability of any particular outcome occurring is skewed.

• The party with the greater knowledge will not necessarily increase his reward from the game.

• One can express this variation as • OPPOSITION >INDIFFERENCE<PREFERENCE• If P1 knows that P2 has a pressing desire to cross the bridge at all

costs he may enforce co-operation but in the sequence P1 before P2. P2 needs to cross the bridge but recognises that getting across the bridge more slowly gives him a better outcome than ending in a DEFECT/DEFECT scenario.

• Unless P2 knows P1’s preference the probable outcome is skewed against the party with the dominant preference and P1’s strategy will dominate.

Prisoners Dilemma –

Tanya and Cinque have been arrested for robbing the Hibernia Savings Bank and placed in separate isolation cells. Both care much more about their personal freedom than about the welfare of their accomplice. A clever prosecutor makes the following offer to each. “You may choose to confess or remain silent. If you confess and your accomplice remains silent I will drop all charges against you and use your testimony to ensure that your accomplice does serious time. Likewise,If your accomplice confesses while you remain silent, they will go free while you do the time. If you both confess I get two convictions, but I'll see to it that you both get early parole. If you both remain silent, I'll have to settle for token sentences on firearms possession charges. If you wish to confess, you must leave a note with the jailer before my return tomorrow morning.”

Rational Self Interest• The “dilemma” faced by the prisoners here is that, whatever

the other does, each is better off confessing than remaining silent. But the outcome obtained when both confess is worse for each than the outcome they would have obtained had both remained silent.

• In a multi-player generalizations model familiar situations in which it is difficult to get rational, selfish agents to cooperate for their common good.

• When multiple games are played with multiple the PD (Prisoners Dilemma) is weakened because groupings can develop where one or more opponents moves to co-operate can be anticipated…The social network develops….

• The “co-operative outcome is generally only obtainable where every player violates rational self interest

The Art of War (Sun Tzu 500BC)

• Only fight a battle you can win.

• Aggression is only a viable option when one has overwhelming force against a vulnerable opponent

• In the absence of overwhelming force manoeuvrability and position is critical.

• Know your opponent, identify his weaknesses, without knowledge you cannot manoeuvre safely.

– Seek knowledge; Even casual conversation will reveal valuable intelligence

– Take the initiative; Remove the opponents key arguments early, often considered the art of deception.

– Plan the surprise; Make a conscious decision when to play a key card

– Gain relative superiority; Making the equal unequal, identify the opponents perceptions and change them

– Be flexible; Accept that as negotiations develop positions change adapt to them change and absorb

Observations• In a single game, the highest probability is that Rational

Self Interest will succeed over Altruism. (the aggressor will dominate the game).

• In a multiple game the party with the greatest demand for a specific outcome is actually WEAKER than the party with no strong dominant preferences.

• Indifference leads to equal probability of highly beneficial outcomes but equally leads to the potential for poor utility returns.

• In multi-game scenarios it is most probable that the parties will weakly dominate on alternate moves and that sequential co-operation will occur leading to less than optimal outcomes for both parties.

• In games with imperfect knowledge and different preferences solving for the optimal strategy may depend on third party forces.

Creating Co-operation Agreements

• In a multiplayer situations contracts will be formed, these may be enduring or may change.

• Example: I an office the partners are trying to decide whether to buy a piece of artwork. You are strongly pro purchase and ranged against you have people with the following preferences.

– Strongly preferential (want to buy)– Highly opinionated and strongly preferential against– Strongly Opposed– Indifferent– Wavering

• Which group of people represent your best allies and why?

Further information

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