Game Theory. Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma. Sequential decisions. Previously …. Sequential decisions with uncertainty Decision trees … with “chance” nodes but …
Game Theory Games of strategy Sequential games Simultaneous decisions Dominated strategies Nash equilibrium Prisoners’ dilemma
Sequential decisions • Previously …. • Sequential decisions with uncertainty • Decision trees … with “chance” nodes • but … • “God does not play dice” – Albert Einstein • “Subtle is the Lord, but malicious He is not.” • What about your competitors?
A sequential “game” • Decisions made in sequence. • Your decision depends on decision made previously by others, and others’ decisions follow and depend on yours, etc. • Outcome/payoff depends on all decisions made by all.
Lucy Van Pelt vs. Charlie Brown • Lucy Van Pelt holds a football on the ground and invites Charlie Brown to run up and kick it. At the last moment, Lucy pulls the ball away. Charles Brown, kicking air, lands on his back, and this gives Lucy great perverse pleasure.
Pull Ball Away Accept Lucy Charlie Let Charlie kick Reject Representing Decisions in a Game Tree , , ,
Games of Strategy Vijay Krishna (Harvard Business School): Any situation where the choices of two or more rational decision makers together leads to gains and losses for them is called a game. A game may simultaneously involve elements of both conflict and co-operation among the decision makers.
Market Competition - HDTV Vizio considers entering a market now monopolised by Samsung. Samsung can decide to respond by being accommodating or aggressively fight a price war. Profit outcomes for both firms depends on the strategies of both firms. As Vizio, you can analyse this problem using Decision Analysis by estimating probabilities of Samsung’s responses.
Market Entry – Decision Tree for Vizio Accommodate $100,000 to Vizio p Samsung Enter 1-p -$200,000 to Vizio Vizio Fight price war Keep out $0 to Vizio How to estimate the probabilities? What does p depend on? If no information, p=0.5? Then Vizio will not enter market.
Game Tree Representation 5, 8 Accommodate Probability of Samsung’s response will depend on Samsung’s payoff in the different scenarios Samsung Enter Market Aggressive Vizio -7, 2 Do not enter 0,10
Market Entry – Game Tree Model $100,000 to Vizio $100,000 to Samsung Accommodate Samsung -$200,000 to Vizio -$100,000 to Samsung Enter Vizio Fight price war Keep out $0 to Vizio $300,000 to Samsung
Analysing Game Trees Rule 1: Look Ahead and Reason Back! • For this market alone, Vizio should choose enter because Samsung (rationally) will accommodate. • If Samsung worries that Vizio may enter other markets in the region after this, Samsung may take a tough stand. Vizio should not enter. • The “payoff” should include all “benefits”.
Look Ahead & Reason Back • Formulate the game tree of the situation. • Identify your own and opponent’s strategy at each stage. This assumes: • Your opponent’s strategy can be observable. • Strategy must be irreversible. • Evaluate payoffs at the “leaves” of the tree. • Think about what will happen at the end. • Reason backward through the tree. • Identify the best strategy for each player at each stage, starting at the end. Note the essence of a game of strategy is interdependence. Your decision affects your opponent’s decision and your opponent’s decision affects yours.
Black-1 Black-1 Black-1 White-2 White-2 White-2 More complex games P-Q4 P-QB4 P-K4 White-1 Theoretically, can map out all possible chess moves and then select the best sequence of moves to win the game!
Chess - Human vs. Computers • Good chess players can “see” 14 moves ahead! • (1968) David Levy: “No computer can beat him in 10 years” • Deep Blue • Chess playing machine built by IBM in the 1990’s • 2 to 2.5 million moves per second. • (1996) Deep Blue 1 lost to world chess champion Gary Kasparov. • (1997) Deep Blue 2 defeated Kasparov.
Homework? • Draw the game tree for TIC-TAC-TOE. • Sure-win strategies?
The Election of the Chief Executive for Hong Kong • The CE of Hong Kong SAR Government Mr. Donald Tsang finishes his term in 2012, and the next CE will be “elected”. • Mr. Henry Tang is Beijing’s favourite candidate. • Mr. Alan Leung (the potential challenger) considers entering the race. • Tang must determine whether to launch a preemptive advertising campaign against Leung (expensive) or not (cost-saving). • Leung must determine whether to enter the race.
Election Game Tree Tang’s, Leung’s payoff In 1, 1 Leung The larger the better Advertise Out 3, 2 Tang In 2, 4 No Ad Leung Out 4, 2
Game Tree Tang’s, Leung’s payoff In 1, 1 Leung Ads Out 3, 2 Tang In 2, 4 No Ads Leung Out 4, 2
Game Tree Tang’s, Leung’s payoff In 1, 1 Leung Ads Out 3, 2 Tang In 2, 4 No Ads Leung Out 4, 2
Advantage due to Order of Decisions? • First-mover advantage? • Tang (first) sets the stage for Leung (second). Tang can look ahead to Leung’s optimal response and make the move to his advantage. • Can Leung improve his situation by acting first?
Game Tree Leung’s, Tang’s payoff Adv 1, 1 Tang In No Adv 4, 2 Leung Adv 2, 3 Out Tang No Adv 2, 4
Better off being first? • Is there a first-mover advantage? • What about adoption of new technology? • Better off as a technology leader? • Better off as a technology follower?
Simultaneous Decisions In the chess example, the sequence of decisions alternate between the players. In other situations, the decision may not be sequential but simultaneous. Tic-tac-toe (sequential) Stone-paper-scissors (simultaneous) In simultaneous games, the payoffs to the players are still interdependent on chosen strategies of ALL players.
Time vs. Newsweek • Each week, these magazines decide on what story to put on the cover. • They do not know the other’s decision until publication. • Suppose there are two “hot” stories: • (A): Anna Chapman, the Russian Spy, • (B): British Petroleum Oil Spill damage • Newsstand buyers only purchase if story is on cover. • 70% interested in (A) and 30% in (B). • Purchases evenly split the if both magazines have the same story.
Matrix Representation of Game What should Time do?
Matrix Representation of Game for Newsweek No matter what Time does, Newsweek is better off putting (A) as cover story.
Dominant Strategies Choosing (A) is a dominant strategy for both Time and Newsweek!
Dominant Strategy • A dominant strategy is one that makes a player better off than he would be if he used any other strategy, no matter what strategy his opponent uses. • A strategy is dominated if there is another strategy that under no circumstances leads to a lower payoff, and sometimes yields a better payoff. • Note: For some games, there may be no dominant strategy for some players.
Properties of a dominant strategy 1: A dominant strategy dominates your other strategies, NOT your opponent! Even with your dominant strategy, your payoff could be smaller than your opponents. 2: A dominant strategy does not requires that the worst possible outcome of the dominant strategy is better than the best outcome of an alternative strategy.
Pricing example Suppose there are just two possible pricing choices: $3 (a profit margin of $2 per copy) and $2 ($1 per copy). Customers will always buy the lower-priced magazine. Profits are split equally between the two. The total readership is 5 million if the price is $3, and rises to 8 million if the price is only $2.
Analysing Games Rule 2: If you have a dominant strategy, use it! Rule 3: Eliminate any dominated strategies from consideration, and do so successively!
C B F E A I D H G Eliminating Dominated Strategies - Example • American ship at A, Iraqi ship at I. • Iraqi plans to fire a missile at American ship; American ship plans to fire a defense missile to neutralize the attack (simultaneously). • Missiles programmed to (possibly) turn every 20 seconds. • If missile not neutralised in 60 seconds, American ship sinks!
Possible strategies (paths) • For American, • A2, A3 dominated by A4, • A6, A7 dominated by A8, • A1 is dominated by A8, • A5 is dominated by A4, • Only A4 and A8 not dominated. • Similarly for Iraqi. • Only I1 and I5 are not dominated.
Simplified Game C B F A E I D H G No dominant strategy for either player!
Nash Equilibrium • A set of strategies constitute a Nash Equilibrium if: no player can benefit by changing her strategy while the other players keep their strategies unchanged. Each player’s strategy is the “best-response” to the other players’ set of strategies.
Dominant Strategy Equilibrium • Higher viewership means more advertising revenues for both TV stations. • Each TV station has a dominant strategy. • In this case, the equilibrium for this game is obvious.
Dominant Strategy Equilibrium • If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.
Nash Equilibrium No dominant strategy for Newsweek. Unique Nash equilibrium.
Example - “Chicken” H > C > D • No dominant strategy • Two Nash equilibria James Dean Swerve Don’t swerve Mad Swerve C, C C, H Max Don’t H, C D, D swerve
Choosing among Multiple Equilibria • Some games have multiple equilibria. • “Rule of the road” • Hong Kong, Britain, Australia, Japan (left) • China, USA (right) • The social convention of the locale determines which equilibrium to choose. Sweden switch from left to right in 1967.
In-class exercise (Texas A&M) • Each of you owns a production plant and can choose to produce 1 or 2 units of a product. • More total production will lower price and hence profit. What would you do?
Is a Nash equilibrium “good” for the players? Just because a game has an equilibrium does not mean that those strategies are “best” for the players. • Prisoners’ dilemma: • Two burglars, Bob and Al, are captured at the scene of a burglary and interrogated separately by the police. • Each has to choose whether or not to confess. • Outcomes: • If neither man confesses, then both will serve only one year. • If both confesses, both will go to prison for 10 years. • However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years.
Prisoners’ dilemma 1950 – Dresher & Flood (Rand) A. W. Tucker • For each player, the dominant strategy is to confess! • Unique Nash equilibrium! • Both play the dominant strategy but create mutually disastrous outcome! Both would be better off by denying!
Cartels • Companies or countries form an alliance to jointly make price and production decisions. • World Trade Organisation (WTO) / General Agreement on Tariffs and Trade (GATT) • The Organization of Petroleum Exporting Countries (OPEC) is a cartel. • the mission of OPEC is to coordinate and unify the policies of its Member Countries and ensure the stabilization of oil markets in order to secure a regular supply of petroleum to consumers, a steady income to producers …
OPEC – Maintaining a Cartel • Total output: 4mb 6mb 8mb • Price per barrel: $25 $15 $10 • Extraction costs: Iran: $2/barrel; Iraq: $4/barrel Dominant strategy: produce at higher level !
Ensuring Co-operation • The dominant strategy equilibrium results in each producing 4 million barrels and achieving 56 million in total joint profit. • Suppose OPEC countries have agreed to maintain production at 2 mb per day. • If members produces 2 million barrels each (as agreed), they will make 88 million in total joint profit. • Is it possible to achieve cooperation, when the dominant strategy is to cheat?
Detection of Cheating • Co-operation is difficult when the reward for cheating is high. • How to tell if some member cheated and produced more? • The price is US$25 per barrel only if members maintained low production. If price drops below $25, then someone has cheated! • What if demand actually decreased?