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# Prisoners dilemma

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### Prisoners dilemma

1. 1. Infinity Business School Personality Enhancement Program (PEP I) Neeraj Batra / Session 01
2. 2. 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) Neeraj Batra / Session 13
3. 3. PEP I Infinity Business School Prisoners Dilemma 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 Neeraj Batra / Session 13
4. 4. PEP I Infinity Business School 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 Neeraj Batra / Session 13
5. 5. PEP I Infinity Business School 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 Neeraj Batra / Session 13
6. 6. PEP I Infinity Business School The Rational Choice 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. Neeraj Batra / Session 13
7. 7. PEP I Outcomes & Strategies Various Infinity Business School 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: 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 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 Neeraj Batra / Session 13
8. 8. PEP I Infinity Business School The Variants of Prisoners Dilemma 1 STAG is PD with Reward reversed CC > DC > DD > CD with Temptation payoff CHICKEN is PD with Punishment 2 DC > CC > CD > DD reversed with Sucker payoff 3 DC > CC > DD > CD This is the Prisoners Dilemma DEADLOCK is PD with Punishment 4 DC > DD > CC > CD reversed with Rewards payoff Neeraj Batra / Session 13
9. 9. PEP I Infinity Business School 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 Neeraj Batra / Session 13
10. 10. PEP I Infinity Business School 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 Neeraj Batra / Session 13
11. 11. PEP I Infinity Business School Common Examples from real life 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. Neeraj Batra / Session 13
12. 12. PEP I Infinity Business School 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. Neeraj Batra / Session 13
13. 13. PEP I Infinity Business School Common Examples from real life 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. Neeraj Batra / Session 13
14. 14. PEP I Infinity Business School Common Examples from real life 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. Neeraj Batra / Session 13
15. 15. PEP I Infinity Business School Prisoners Dilemma 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: Neeraj Batra / Session 13
16. 16. PEP I Infinity Business School Prisoners Dilemma Business Example – Zero Sum Game 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. Neeraj Batra / Session 13
17. 17. PEP I Infinity Business School Prisoners Dilemma Business Example – Zero Sum Game Here is the payoff table for these two companies Payoff Table Pepsi Price Rs.10 Rs.20 Rs.10 0,0 5 Mn, -5Mn Coca cola Rs.20 -5 Mn, 5 Mn 0,0 Neeraj Batra / Session 13
18. 18. PEP I Infinity Business School Prisoners Dilemma Business Example – Zero Sum Game 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. Neeraj Batra / Session 13
19. 19. PEP I Infinity Business School 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. Table (Payoff in Mns) Pure Life Bottles P=1 P=2 P=3 P=1 0,0 50,-10 40, -20 Green P=2 -10,50 20, 20 90, 10 Bottles P=3 -20,40 10, 90 50, 50 Neeraj Batra / Session 13
20. 20. PEP I Infinity Business School Prisoners Dilemma 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. Neeraj Batra / Session 13
21. 21. PEP I Infinity Business School 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: Table (Market Shares) KOOL LOW MED HI LOW 20, 25 50, 30 50, 20 LAYS MED 30, 50 15, 15 30, 20 HI 20, 50 20, 30 10, 10 Neeraj Batra / Session 13
22. 22. PEP I Infinity Business School Prisoners Dilemma Games with Multiple Nash Equilibria 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 Neeraj Batra / Session 13
23. 23. PEP I Infinity Business School Prisoners Dilemma Games with Multiple Nash Equilibria 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. Neeraj Batra / Session 13
24. 24. PEP I Infinity Business School Prisoners Dilemma Games with Multiple Nash Equilibria 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! Neeraj Batra / Session 13
25. 25. PEP I Infinity Business School 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. Neeraj Batra / Session 13
26. 26. PEP I Infinity Business School Prisoners Dilemma Multiple Players: The Queuing Game 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. Neeraj Batra / Session 13
27. 27. PEP I Infinity Business School Prisoners Dilemma Multiple Players: The Queuing Game 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. 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 Neeraj Batra / Session 13
28. 28. PEP I Infinity Business School Prisoners Dilemma Multiple Players: The Queuing Game 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. Neeraj Batra / Session 13
29. 29. PEP I Infinity Business School Prisoners Dilemma Multiple Players: The Queuing Game 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. This leaves the average payoff at 10 for those who remain. Since the 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. This leaves the average payoff at 9 for those who remain. Since the 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. Neeraj Batra / Session 13
30. 30. PEP I Infinity Business School Prisoners Dilemma Multiple Players: The Queuing Game 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. Neeraj Batra / Session 13
31. 31. PEP I Infinity Business School Prisoners Dilemma Multiple Players: The Queuing Game 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. Neeraj Batra / Session 13
32. 32. PEP I Infinity Business School 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.: 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) Neeraj Batra / Session 13
33. 33. PEP I Infinity Business School Prisoners Dilemma The Marketing Game 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 Neeraj Batra / Session 13
34. 34. PEP I Infinity Business School Prisoners Dilemma Some other points 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 Neeraj Batra / Session 13
35. 35. PEP I Infinity Business School Prisoners Dilemma Some other points 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 Neeraj Batra / Session 13
36. 36. PEP I Infinity Business School Prisoners Dilemma Some other points 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 Neeraj Batra / Session 13
37. 37. PEP I Infinity Business School Prisoners Dilemma THE FOUR RULES 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?) Neeraj Batra / Session 13
38. 38. PEP I Infinity Business School Prisoners Dilemma THE FOUR RULES 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) Neeraj Batra / Session 13
39. 39. PEP I Infinity Business School Prisoners Dilemma The Four Rules 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 ? Neeraj Batra / Session 13
40. 40. PEP I PEP I Infinity Business School Infinity Business School Prisoners Dilemma The Four Rules 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 Neeraj Batra / Session 13