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1. Game Theory Let’s play a game, the centipede game A B A B A B R r R r R r 5.00 5.00 D d D d D d 1 1 0 3 2.50 2.50 1.50 4.50 3.50 3.50 3.00 6.00

2. Game Theory Strategy and Strategic Decision Making Objective: Better decision making Approach: Strategy Strategic decision making is characterized by interactive pay off. Interactive payoff means that the outcome of your decision depends on both your actions and the action of others.

3. Rules to Live (or at least play) By Rule No 1: Understand the rules Rule No 0: Life is a game What are the basic rules of the game? 1. There are players (stakeholders) in every game (situation). Know as many of them as possible. 2. Know what can and what cannot be done. Identify the feasible strategy set 3. Feasible strategy sets of players interact to form a set of possible outcomes. Know what can and what cannot happen. Know all (or as many as possible of) potential outcomes.

4. 4. Know the rewards and the punishments. Each outcome has a payoff. Know each payoff. 5. Players are rational but preferences are subjective. 6. Know the order of the play. Games can be simultaneous or sequential. 7. Know the extent of the game. Games can be one-shots, finite horizon repeating or indefinite repeating.

5. Example: as it applies to business decision-making Dewey, Cheetham and Howe Inc. (DC&H) and Rupf & Reddy Inc (RR) are two competing firms. DC&H and R&R both wish to launch what would essentially be competing products. Each has the option of either keeping their development spending at current rates or to escalate so it can be first to market. There are no spies, neither knows about the other’s decision until put into effect In a rare stroke of luck, the management of both companies are sane

6. If DC&H spends at the current level when R&R escalates, then DC&H will make $3 Mil and R&R will make $2 Mil. If DC&H spends at the current level and R&R stays, then DC&H will make $3 Mil and R&R will make $4Mil If DC&H escalates and R&R stays, then DC&H will make $4 Mil and R&R will make $3 Mil If DC&H escalates and R&R also escalates, then DC&H will make $3 Mil and R&R will make $2 Mil Now let us identify the central elements:

7. Players – DC&H and R&R • Each player has the choice to either stay (spend at current levels) or escalate (put more money into the project) • There are four outcomes: when both escalate, when both stay and two cases when one escalates and one stays. • Payoffs are as reflected in the table below: • We assume that decisions will be taken such that to maximize payoff to self. • We won’t know of the other player’s move so the game is simultaneous • Once the decision is made, that’s it, so the game is a one-shot. Game (Strategy) Matrix

8. Dominant Strategy Should DC&H stay or escalate? How about R&R? Game (Strategy) Matrix A dominant strategy is one whose payoff in any outcome, relative to all other feasible strategies, is the highest.

9. Dominant Strategy The dominant strategy is therefore to first look for a dominant strategy How to: identify and remove all dominated strategies What is left is either a dominant strategy or a field of options that are clearly not dominated. Sometimes, removing a dominated strategy would change a previously non-dominated strategy into a dominated one and as such a candidate for removal

10. Example: DC&H and R&R must now decide on a pricing policy for the new product. They know now that the other party will introduce a new and similar product. R&R will have three pricing options $1.65, $1.35 and $1.00 DC&H will also have three pricing options, $1.55, $1.30, $0.95 The payoffs are as below: Game (Strategy) Matrix

11. Game (Strategy) Matrix R&R will charge $1.35, and DC&H will charge $1.55 At this stage, no party needs to unilaterally change strategy, we have reached A DOMINANT STRATEGY EQUILIBRIUM

12. Nash’s Equilibrium Not many games settle in a dominant strategy equilibrium. This is because not all games have a clear dominant strategy. Those that do are called DOMINANCE SOLVABLE. How do we predict behavior in a game without dominant strategies. We need to include the future (anticipated) rational actions of others and still arrive at a rational, stable and optimal solution. We reach Nash Equilibrium when all players choose their best strategy assuming that their rivals have done or will do likewise.

13. Note: • This does not mean that the game players will cooperate with each other. It simply means that they will do the best for themselves knowing that the competition is doing the same. • The essence of success becomes correctly predicting the decisions of others. • Only a Nash equilibrium pair (or set) will be optimum for both (all) players • There may be more than one Nash equilibrium point.

14. Example: In the matrix below we have R&R and DC&H’s profit figures. We assume that R&R entered the market first and that both firms wish to introduce new products. Also that each can choose amongst several but must settle on only one product. What product they will choose depends on what the competition will do. The outcomes and payoffs are captured below:

15. A simple check would indicate that there is no dominated strategy for either firm. For each strategy, we indicate the behavior of others: For example if D&CH knew that R&R will introduce product A, what will they do? DC&H would introduce Product 3, as it gives the highest payoff.

16. D D D if R&R would introduce product B, then DC&H will introduce Product 1 Likewise if R&R would introduce Product C, DC&H will introduce Product 2

17. Game (Strategy) Matrix R D R D D R Now doing the same analysis this time for DC&H: The Nash equilibrium pair is when R&R introduces Product A and D&CH introduces product 3.

18. Strategic Foresight Successful game players often find that they need to make decisions now that would be rational if what is anticipated actually happens in the future This is called Strategic Foresight Game theory can formally model strategic foresight through the process of backward induction. Backward induction is using future information to move backward in time (sequence) to arrive at a logical situation in the present. However, we do need to present an alternate form of game information presentation to best utilize this approach

19. The Extensive Form Game information may also be presented using what is termed a “game tree”. Using the pricing information relative to DC&H and R&R, as presented before we can also present the game information as below. 8,5 $1.55 DC&H $1.30 6,2 $0.95 6,0 $1.65 14,7 $1.55 DC&H $1.35 $1.30 8,2 R&R 5,1 $0.95 $1.00 DC&H 10,4 $1.55 $1.30 7,1 $0.95 3,6

20. Backward Induction For instance R&R and DC&H wish to decide whether to expand or not. The game tree with payoff is illustrated below 80,80 Do not Expand DC&H Expand 60,120 Do not Expand R&R Expand DC&H Do not Expand 150,60 Expand 50,50 Let us solve this game using backward induction

21. D A. If R&R expands, then DC&H will receive $50 mil if they expand 80,80 Do not Expand DC&H B. If R&R expands, then DC&H will receive $60 mil if they don’t expand Expand 60,120 Do not Expand So R&R managers anticipate that if they expand, DC&H will not C R&R Expand DC&H Do not Expand 150,60 Expand B 50,50 So R&R managers anticipate that if they do not expand, then DC&H will A C. If R&R do not expand, then DC&H will receive $120 mil if they expand D. If R&R do not expand, then DC&H will receive $80 mil if they don’t expand So if R&R expands, they anticipate a $150 mil payoff (because R&R will not expand), and if they do not expand, the payoff is $60 mil, so R&R will expand Given that R&R will expand, DC&H will not

22. Example: The Centipede Game A B A B A B R r R r R r 5.00 5.00 D d D d D d 1 1 0 3 2.50 2.50 1.50 4.50 3.50 3.50 3.00 6.00 Now use backwards induction to solve the game

23. Threats, Commitments and Credibility A kiss on the hand is very continental but diamonds are a girl’s best friend Should you believe others? When should you believe others? How do you test for credibility? A major use of backwards induction is to test out the credibility of threats or commitments of your opponents. Another dominant strategy is to ALWAYS check for credibility first Only consider credible commitments Always? Well , almost always

24. Consider the following situation: R&R have expanded the product line and now DC&H wish to counter by dropping the price of their product. However they are concerned that if they dropped the price, R&R would also drop theirs. R&R are telling some common suppliers that they would drop their price if DC&H would. The tree below depicts the situation and the payoffs to each (values in $Million) Maintain Price 50,30 R&R Drop Price 70,20 Maintain Price DC&H Drop Price R&R Maintain Price 30,40 Drop Price 20,15 Is this threat credible?

25. Maintain Price 50,30 R&R Drop Price 70,20 Maintain Price DC&H Drop Price R&R Maintain Price 30,40 Drop Price 20,15 If DC&H drop prices, R&R will maintain theirs (otherwise they would lose $10mil). If DC&H maintains prices, R&R will drop theirs (otherwise they would lose $20mil). DC&H should drop prices as the R&R threat is not credible

26. Price of Distrust Game (Strategy) Matrix Consider the consequence of pricing policies per sales period Playing this game once, would have both parties price low (as they cannot afford both to price high and see the competition drop their prices). As such they each lose $2 mil per period. They could increase their respective profits by $2 mil each if they could trust each other When you are playing the game once, there is no reason to trust, but if you are in it for the long haul, the situation is different

27. Repeated Games Once there is the prospect of a future, behavior changes as new concepts such as trust, reputation, reciprocity and revenge come into play. Remember: Commitments must remain credible Indefinite Games: Games that continue without any knowledge of whether they will terminate or when they will do so. Definite Games: Games that continue for a time but is known to end at a particular instance. Under which circumstance is it easier to establish and maintain trust?

28. Definite (finite horizon) Games Game (Strategy) Matrix Consider again the pricing policies per sales period of DC&H and R&R If the two firms cooperate and price high each receives a payoff of $5 mil. If one defects, and prices low, it will have a windfall of $20 mil for a single period. The other will then price low and each will receive $3 mil. So the incremental of $15 mil will be more than eroded in 8 cycles. In finite horizon games, as the game progresses, the impact and importance of the future shrinks. In the last period, the Nash equilibrium is identical to a one-shot game. Using backward induction one can see that the equilibrium for the entire game will be forced into one identical to a one-shot.

29. Indefinite (infinite horizon) Games These are fundamentally different. The equilibrium becomes a function of probable future behavior. These are of course much harder to predict. The presence of a future and incomplete information are the necessary ingredients for building reputation Reputation is simply the integrated history of past behavior. The past is a good indicator of the future.

30. Cheating Many of us have cheated in mathematics, let us look at the mathematics of cheating! Imagine a one-shot game: PN Nash equilibrium profit PC Profit when cooperating PH Profit when cheating In this game there are three possible profit levels: We can calculate the benefit from cheating as: B=PH-PC As presumably we cheat to get an advantage, B should always be positive. We can calculate the cost of cheating as: C=Pc-PN But as the game is one-shot and there is no consequence to cheating C is always Zero. Absolute profit is Π=B-C=PH-PC – 0 = PH-PC which is always positive

31. Now Imagine a definite repeating game: The present value of multiple rounds of such game is: Rule: Not cheating maximizes the value of a firm when the present value of the costs of cheating is greater than the present value of the benefits from cheating.

32. Finally, imagine an indefinite repeating game: The present value of multiple rounds of such game is: Rule: Not cheating always maximizes the value of a firm because at some stage (N) cheating will be discovered and from there on there will be a cost that eventually will become greater than the value of benefits from cheating (Provided of course that the game will go on for long enough)

33. Coordination Games When a game has more than one Nash equilibrium, any one such equilibrium might be selected by a given player. By coordination they may be able to improve their odds by selecting the most preferred equilibrium point. There are many different types of coordination some collaborative, some competitive. We shall investigate some here.

34. Matching Games In this game, players generally have the same preferences in the outcome they seek. Impediment may be in ability to communicate or asymmetric information. In the game below, both 7,7 and 12,12 are Nash equilibria. But both DC&H and R&R would no doubt prefer 12,12. With collaboration and communication (and ensuring that neither party will cheat, both parties can settle on a coordinated strategy of DC&H producing for the consumer market and R&R for the industial market. Game (Strategy) Matrix

35. Battle of Sexes In this game players still wish to coordinate but on different outcomes. Each preferred payoff by one is NOT favored by the other. If the game is repeated indefinitely, players often switch between outcomes so both would gain. In one shot outcomes, it is impossible to predict the outcome without good knowledge of the players reputations and styles. Game (Strategy) Matrix

36. Hawks and Doves In this game the players are locked in a conflict. If both act like hawks, there is usually poor payoff (often loss) as a consequence of conflict. If one acts hawkish and the other dovish, the hawk has an immediate advantage Game (Strategy) Matrix