Prisoners dilemma

Neeraj Batra / Session 01

Personality Enhancement Program

(PEP I)

Infinity Business School

PEP I Infinity Business School


Prisoners Dilemma

Imagine Angie & Bert Caught In A Theft

Separated By The Police To Seek Confession

The Police Motivate Defection With Leniency

The Theoretical Options Can Be:

They both co-operate with each other (i.e don’t confess)

This is referred as (C,C)

One of them co-operates,the other defects (i.e rats on the other)

This is referred as (C,D)

They both rat on each other, i.e they both defect

This is referred as (D,D)


If The Punishment Matrix (also called the Payoff) is as follows:

DC (0,10) means the first guy gets 0 years,the second guy 10 years

CC (1,1) means each of them gets 1 year

DD (4,4) means each of them gets 4 years

CD (10,0) means the first guy gets 10 years,the second guy 0 years

This is referred to as the Payoff Matrix

In Order of Preference For Each Of Them DC > CC > DD > CD

Defection Is Thus A Dominant Strategy in The Prisoners Dilemma

In A Symmetrical Game the Equilibrium Lies at DD

Nash's Equilibrium


PEP IPrisoners Dilemma


Key Terminology in Prisoners Dilemma

DC is called the Temptation Payoff : The payoff for taking the

temptation to Defect on your partner

CC is called the Reward or Mutual Payoff : The payoff for co-

operating with your partner

DD is called the Punishment Payoff : The payoff for defecting on your

partner who defects similarly

CD is called the Suckers Payoff : The payoff for cooperating when

your partner is defecting

Symmetric Games are games in which the order of the players

action does not cause any dynamic change in the payoff matrix



Prisoners Dilemma

Lets Look At The Payoff Matrix For Angie & Bert

Option 1 Option 2 Option 3 Option 4

Angie C D C D

Bert C D D C

Angies Punishment (Payoff) 1 4 10 0

Berts Punishment (Payoff) 1 4 0 10

Total Punishment (Payoff) 2 8 10 10



In the traditional prisoners dilemma, the exasperating conclusion

any rational prisoner faces is that there is really no choice but to

defect. Considering what the other person might do, for each case,

your best option (less time in jail) is to defect. Of course, your

partner comes to the same conclusion. The net result is a situation

that is inferior to the situation you would get if both cooperated.

The Rational Choice



Theoretically you can draw up 24 Payoff Matrices of various ranking

combinations for a 2 x 2 game.

However only 6 Payoff Matrices would make logical sense to exist. These

are as follows:

Out of the above, only 3,4,5 & 6 qualify for any strategy in which

Defection gives better results. Thus either DC > CC or DD > CD or both

must happen otherwise there is no incentive to defect.

Clearly Payoff matrices 1 and 2 are out in this case

Various Outcomes & Strategies

1 CC > CD > DC > DD Co-operation is better

2 CC > DC > CD > DD Co-operation is better

3 CC > DC > DD > CD Tit for Tat works : Stag Hunt Strategy

4 DC > CC > CD > DD Reverse Strategy is better: Chicken

5 DC > CC > DD > CD Defection Works: Prisoners Dilemma

6 DC > DD > CC > CD Defection works best: Deadlock Strategy



The Variants of Prisoners Dilemma


1CC > DC > DD > CD

STAG is PD with Reward reversed

with Temptation payoff

2 DC > CC > CD > DDCHICKEN is PD with Punishment

reversed with Sucker payoff

3 DC > CC > DD > CD This is the Prisoners Dilemma

4 DC > DD > CC > CDDEADLOCK is PD with Punishment

reversed with Rewards payoff


The Chicken Strategy


Angie and Bert are driving cars

Coming from opposite directions

One who swerves first is the chicken.

Best payoff is:

DC > CC > CD > DD


The Stag Strategy


There are two hunters & 2 games:

A RABBIT AND A DEER.

Chances of getting the deer independently:

NEXT TO ZERO

The deer meat will be shared :

75:25 in favour of one who snare the deer

CO-OPERATION 50:50 TO EACH.

Best payoff is

CC > DC > DD > CD


The Fire:

A club has a fire, and all rush for the exits, preventing the exit of

anyone; as a result, all perish.

The Concert:

At a concert, Amit stands on his toes to see the performer better. The

person behind Amit is forced to stand, and the effect ripples

throughout the auditorium. Soon all are standing, and no one has a

better view than they would have had in a sitting position, except that

now they must stand versus sit.

Common Examples from real life



Ragging:

No one likes being ragged. You get butterflies when you visit

college on the first day. However you were ragged as a fresher.

Thus, you must act the senior for retribution and to maintain

“tradition” and ragging continues indefinitely.

Steroids:

Athlete A uses steroids, which gives him a competitive advantage.

Other athletes are forced to use steroids to retain parity. As a result,

no athlete is given a competitive advantage, but all are subjected to

the hazards of steroids.




Free Email:

A leading provider of email services starts providing free email to

gain market share and very soon so does everyone else leaving

neither with a competitive advantage. Soon all email service

providers go belly up.

Cell Phone Operators:

A cell phone operator cuts the pulse rate to pull new subscribers,

almost thereafter so does all his competitors thereby bringing all rates

down and poaching each others clients. Ultimately each company

swaps each others clients but none are better off in number than

before leaving them greatly poorer.




Celebrity Endorsements:

A leading soft drink manufacturer uses the heart throb hero to promote

their cola over a competitor, very soon the competitor gets another hero

to do the same. The heroes are paid Rs 10 Mn each leaving both

companies that much poorer.




Business Example – Zero Sum Game

Here is an example of a zero-sum game. It is a very simplified

model of price competition. Assume two Cola companies have a

fixed cost of Rs 5 Mn per period, regardless whether they sell

anything or not. We will call the companies Pepsi and Coke, just to

take two names at random.

The two companies are competing for the same market and each

firm must choose a high price (Rs 20 per bottle) or a low price

(Rs10 per bottle). Here are the rules of the game:

Prisoners Dilemma




1) At a price of Rs20, 500,000 bottles can be sold for a total revenue of Rs10

Mn

2) At a price of Rs10, 1,000,000 bottles can be sold for a total revenue of Rs

10 Mn

3) If both companies charge the same price, they split the sales evenly

between them.

4) If one company charges a higher price, the company with the lower price

sells the whole amount and the company with the higher price sells

nothing.

5) Payoffs are profits -- revenue minus the Rs 5 Mn fixed cost.

Prisoners Dilemma




Here is the payoff table for these two companies

Prisoners Dilemma

Pepsi

Price Rs.10 Rs.20

Coca colaRs.10 0,0 5 Mn, -5Mn

Rs.20 -5 Mn, 5 Mn 0,0

Payoff Table




This is a zero-sum game.For two-person zero-sum games, there is a

clear concept of a solution. The solution to the game is the maximum

criterion -- that is, each player chooses the strategy that maximizes her

minimum payoff. In this game, Coke’s minimum payoff at a price of

Rs10 is zero, and at a price of Rs20 it is –5 Mn, so the Rs10 price

maximizes the minimum payoff. The same reasoning applies to Pepsi,

so both will choose the Rs10 price.

Here is the reasoning behind the maximum solution: Coke knows that

whatever it loses, Pepsi gains; so whatever strategy it chooses, Pepsi

will choose the strategy that gives the minimum payoff for that row.

Again, Pepsi reasons conversely.

Prisoners Dilemma



Price Competition Example

Think of two companies selling "bottles" at a price of one, two, or three rupees

per bottle. The payoffs are profits -- after allowing for costs of all kinds -- and

are shown in the table below. The general idea behind the example is that the

company that charges a lower price will get more customers and thus, within

limits, more profits than the high-price competitor.

Prisoners Dilemma

Pure Life Bottles

P=1 P=2 P=3

Green

Bottles

P=1 0,0 50,-10 40, -20

P=2 -10,50 20, 20 90, 10

P=3 -20,40 10, 90 50, 50

Table (Payoff in Mns)



Price Competition Example

We can see that this is not a zero-sum game. Profits may add up to

100, 20, 40, or zero, depending on the strategies that the two

competitors choose. We can also see fairly easily that there is no

dominant strategy equilibrium.

Green Bottles company can reason as follows: if Pure Life Bottles

were to choose a price of 3, then its best competitive price is 2, but

otherwise Green Bottles best price is 1 , thus there is no dominant

strategy.

Prisoners Dilemma



Games with Multiple Nash Equilibria

Two crisp manufacturers (LAYS and KOOL) have to choose prices for their

packs. There are three possible prices: Rs 8 (LOW), Rs 10 (MED) or Rs 12 (HI).

The audiences for the three prices are 50 Mn, 30 Mn, and 20 Mn, respectively. If

they choose the same prices they will split the audience for that market equally,

while if they choose different prices, each will get the total customer base for

that price band. Market shares are proportionate to payoffs. The payoffs (market

shares) are in the Table below:

Prisoners Dilemma

KOOL

LOW MED HI

LAYS

LOW 20, 25 50, 30 50, 20

MED 30, 50 15, 15 30, 20

HI 20, 50 20, 30 10, 10

Table (Market Shares)




You should be able to verify that this is a non-constant sum game, and that

there are no dominant strategy equilibria. If we find the Nash Equilibria by

elimination, we find that there are two of them -- the upper middle cell and

the middle-left one, in both of which one station chooses LOW and gets a

50 market share and the other chooses MED and gets 30 (Hint: These have

the highest total payoff). But it doesn't matter which company chooses

which format.

It may seem that this makes little difference, since

• the total payoff is the same in both cases, namely 80

• both are efficient, in that there is no larger total payoff than 80

Prisoners Dilemma




There are multiple Nash Equilibria in which neither of these things

is so. But even when they are both true, the multiplication of

equilibria creates a danger. The danger is that both stations will

choose the more profitable LOW format -- and split the market,

getting only 25 each! Actually, there is an even worse danger that

each station might assume that the other station will choose LOW,

and each choose MID, splitting that market and leaving each with a

market share of just 15.

Prisoners Dilemma




More generally, the problem for the players is to figure out which

equilibrium will in fact occur. In still other words, a game of this

kind raises a "coordination problem:" how can the two companies

coordinate their choices of strategies and avoid the danger of a

mutually inferior outcome such as splitting the market? Games

that present coordination problems are sometimes called

coordination games.

From this point of view, we might say that multiple Nash

equilibria provide us with a possible "explanation" of coordination

problems. That would be an important positive finding, not a

problem!

Prisoners Dilemma



Multiple Players: The Queuing Game

Many of the "games" that are most important in the real world

involve considerably more than two players.

In this sort of model, we assume that all players are identical,

have the same strategy options and get symmetrical payoffs. We

also assume that the payoff to each player depends only on the

number of other players who choose each strategy, and not on

which agent chooses which strategy.

Prisoners Dilemma




As usual, let us begin with a story. We suppose that six people are

waiting at the ration queue, but that the clerks have not yet arrived at

the counter to serve them. Anyway, they are sitting and awaiting their

chance to be called, and one of them stands up and steps to the office

to bribe and be the first in the queue. As a result the others feel that

they, too, must bribe to get ahead in the queue, and a number of people

end up bribing when they could all have been honest.

Prisoners Dilemma




Here is a numerical example to illustrate a payoff structure that might lead to

this result. Let us suppose that there are six people, and that the gross payoff

to each passenger depends on when she is served, with payoffs as follows in

the second column of Table X. Order of service is listed in the first column.

Prisoners Dilemma

Table X

Order served Gross Payoff Net Payoff

First 20 18

Second 15 13

Third 13 11

Fourth 10 8

Fifth 9 7

Sixth 8 6




The Gross payoffs are however before the bribe. There is a two-point

bribe reduction for a participant. These net payoffs are given in the

third column of the table.

Those who do not bribe are chosen for service at random, after those

who stand in line have been served. If no-one bribes, then each person

has an equal chance of being served first, second, ..., sixth, and an

expected payoff of 12.50 In such a case the aggregate payoff is 75.

But this will not be the case, since an individual can improve her

payoff by bribing, provided she is first in line. The net payoff to the

person first in line is 18 >12.5, so someone will get up and bribe.

Prisoners Dilemma




This leaves the average payoff at 11 for those who remain. Since the

second person in line gets a net payoff of 13, someone will be better off

to get up and bribe for the second place in line.


third person in line gets a net payoff of 11, someone will be better off

to get up and bribe for the third place in line.


fourth person in line gets a net payoff of 8, which is less than the

payoff of 9, there is no further incentive for anyone to bribe the clerks

and we reach the game’s Nash’s equilibrium.

Prisoners Dilemma




The total payoff is 69, far less than the 75 that would have been the

total payoff if, somehow, the bribing could have been prevented.

Two people are better off -- the first two in line -- with the first gaining

an assured payoff of 5.5 above the uncertain average payoff she would

have had in the absence of queuing and the second gaining 0.5. But the

rest are worse off. The third person in line gets 12, losing 1.5; and the

rest get average payoffs of 9, losing 3.5 each. Since the total gains

from bribing are 6 and the losses 12, we can say that, in one fairly

clear sense, bribing is inefficient.

Prisoners Dilemma




We should note that it is in the power of the authority (the ration, in this

case) to prevent this inefficiency by the simple expedient of not

respecting the bribe and increasing transparency of queuing. If the

clerks were to ignore the bribe and, let us say, pass out lots for order of

service at the time of their arrival, there would be no point for anybody

to bribe, and there would be no effort wasted by queuing (in an

equilibrial information state).

This is a representation of an asymmetrical model of Prisoners

dilemma.

Prisoners Dilemma



The Marketing Game

4 Companies play for market share by Co-operating on pricing i.e. Do

Not Undercut Competitors Vs Defecting Undercut Competitors. The

Payoff Matrix for the Game is as follows.:

Prisoners Dilemma

Round 01 :

(4Cs 12,12,12,12)

(2Cs, 2Ds 10,15)

(4Ds 9, 9, 9, 9)

(3Cs, D 10,10,10,20)

(C, 3Ds 10,12)



The Marketing Game

Prisoners Dilemma

Round 03 :

(4Cs 12,12,12,12)

(2Cs, 2Ds 11,11)

(4Ds 11,11,11,11)

(3Cs, D 11,10)

(C, 3Ds 12,10)

Round 05 :

(C & D Combination) 12,14

All Cs 13 each

All Ds 11



Some other points

Prisoners Dilemma

A+B =A+B+A’B’ (Synergy Effect)

Example Wolves Pack : Hunting/Opec

Nash’s Equilibrium is the point where neither player can

unilaterally improve his position any further.

In most one round games DD is the Nash’s Equilibrium



Some other points

Prisoners Dilemma

To increase rounds or “lengthen the shadow of future” there are several

techniques used such as : Arms control (phased destructions)

To get Co-operation phased extension of country loans by IMF

In intense cardinal payoffs co-operation can be given huge payoffs e.g. if

China adheres to the WTO guidelines it gets an MFN status

Anti-dumping duties is another effective way of discouraging defection



Some other points

Prisoners Dilemma

Improving communication and information exchange can increase co-

operation

By reducing transaction costs/inefficiencies co-operation can be

institutionalized (Payoff matrix) History plays a key role in the key

players strategy.

Trust plays a key role in key players strategy

The payoff matrix plays a key role in key players strategy



THE FOUR RULES

Prisoners Dilemma

Rule No 1 : THE GOLDEN RULE

DO UNTO OTHERS AS YOU WOULD HAVE

THEM DO UNTO YOU

(Repay Even Evil With Forgiveness- Jesus Christ)

(BUT If You Repay Evil With Kindness, With What

Will You Repay Kindness?)



THE FOUR RULES

Prisoners Dilemma

Rule No. 2: THE SILVER RULE


DO UNTO OTHERS WHAT YOU WOULD NOT

HAVE THEM DO UNTO YOU

(Martin King Luther/Mahatma Gandhi)

(Principle of Non Co-operation)


The Four Rules

Prisoners Dilemma

Rule No. 3: THE BRONZE RULE

DO UNTO OTHERS AS THEY DO UNTO YOU

(Repay Kindness With Kindness, Evil With Justice -

Confucius)

LEX TALIONIS PRINCIPLE

What Happens To Two Wrongs Don’t Make A Right ?


PEP IInfinity Business School

The Four Rules

Prisoners Dilemma

Rule No. 4 : THE IRON RULE

DO UNTO OTHERS AS YOU LIKE BEFORE THEY DO

UNTO YOU

(The Power Rule, Provided You Can Get Away With It)

Leads to Inconsistency : Suck Up to those Above, Exploit

Those Below


PEP IPEP IPEP IPrisoners Dilemma
