1. Mobile Computing Group An Introduction to Game Theory part II Vangelis Angelakis 21 / 10 / 2004

2. Previously on Part I… TNL - Mobile Computing Group Angelakis Vangelis 21/10/2004 Game theory is the mathematically founded study of conflict and cooperation. In non-cooperative games cooperation arises, when it is in the best interest of players Rational players do not play dominated strategies. A strategic form game can be a single round of a repeated game Players are in a Nash equilibrium if a unilateral change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. Under mixed strategies any game in strategic-form has an equilibrium Game theory is the mathematically founded study of conflict and cooperation. In non-cooperative games cooperation arises, when it is in the best interest of players Rational players do not play dominated strategies. A strategic form game can be a single round of a repeated game Players are in a Nash equilibrium if a unilateral change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. Under mixed strategies any game in strategic-form has an equilibrium

3. Extensive games TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Games in strategic form have no temporal component. Players choose their strategies simultaneously not knowing how other players moved. An extensive form game or game tree is a more detailed model than a normal form game. It represents a formalization of interactions where players can over time be informed about the actions of others. Each player moves at a designated time & order and no simultaneous moves are allowed. In an extensive game of perfect information each player is aware of all the previous choices of all other players. Games in strategic form have no temporal component. Players choose their strategies simultaneously not knowing how other players moved. An extensive form game or game tree is a more detailed model than a normal form game. It represents a formalization of interactions where players can over time be informed about the actions of others. Each player moves at a designated time & order and no simultaneous moves are allowed. In an extensive game of perfect information each player is aware of all the previous choices of all other players.

4. Remember this: Quality Selection TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Customer VoIP Provider BuyDon’t buy High 2222 1010 Low 0303 1111 1

5. A third variant to the same game TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 VoIP Provider Customer High Low Buy Don’t buy Buy ( 2, 2 ) ( 0, 1 ) ( 3, 0 ) ( 1, 1 ) Don’t buy Assume the VoIP provider moves first, say by announcing the quality level he will provide and commits himself to that move

6. Backward Induction TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 VoIP Provider Customer High Low Buy Don’t buy Buy ( 2, 2 ) ( 0, 1 ) ( 3, 0 ) ( 1, 1 ) Don’t buy Unique solution! Backward induction always defines at least one Nash equilibrium. Backward induction usually prescribes unique choices at the players’ decision nodes.

7. Strategies in extensive games TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Backward induction gives a complete plan of what to do at each point in the game when the player can make a move, even though that point may never arise in the course of a game Such a plan is called a strategy of a player. Eg. Backward induction gives a complete plan of what to do at each point in the game when the player can make a move, even though that point may never arise in the course of a game Such a plan is called a strategy of a player. Eg. VoIP Provider High Low Buy Don’t buy Buy ( 2, 2 ) ( 0, 1 ) ( 3, 0 ) ( 1, 1 ) Don’t buy A Customer strategy would be: Buy if High, Don’t buy if Low Where in a rational context, only the first will come into effect. A Customer strategy would be: Buy if High, Don’t buy if Low Where in a rational context, only the first will come into effect.

8. Strategic forms of extensive games TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 VoIP Provider High Low Buy Don’t buy Buy ( 2, 2 ) ( 0, 1 ) ( 3, 0 ) ( 1, 1 ) Don’t buy Customer VoIP Provider Buy/High Buy/Low Buy/High Don’t/Low Don’t/High Buy/Low Don’t /High Don’t/Low High 2222 2222 1010 1010 Low 0303 1111 0303 1111 The strategic form of a game tree tabulates all possible strategies of the players All move combinations of the second player must be distinguished as strategies since any two of them could lead to different results depending on the moves of player 1. The strategic form of a game tree tabulates all possible strategies of the players All move combinations of the second player must be distinguished as strategies since any two of them could lead to different results depending on the moves of player 1.

9. Nash equilibria in extensive games TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Backward induction always defines a Nash equilibrium But not all possible Nash equilibria of an extensive game arise by backward induction. Backward induction always defines a Nash equilibrium But not all possible Nash equilibria of an extensive game arise by backward induction. Customer VoIP Provider Buy/High Buy/Low Buy/High Don’t/Low Don’t/High Buy/Low Don’t /High Don’t/Low High 2222 2222 1010 1010 Low 0303 1111 0303 1111 Don’t/High, Don’t/Low Is a Nash equilibrium, Given the knowledge that player 1 has chosen low. This is a suboptimal solution to a subgame. Backward induction derived Nash equilibria are subgame perfect Don’t/High, Don’t/Low Is a Nash equilibrium, Given the knowledge that player 1 has chosen low. This is a suboptimal solution to a subgame. Backward induction derived Nash equilibria are subgame perfect

10. First mover advantage TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Many games in strategic form exhibit what is called first-mover advantage A player in a game becomes the first mover or leader when he can commit to a strategy and inform other players about it. First mover advantage states that a player who can become a leader will not be worse off than an original game where all players act simultaneously. Also known as Stackelberg leadership If a player has the power to commit he should do so. Many games in strategic form exhibit what is called first-mover advantage A player in a game becomes the first mover or leader when he can commit to a strategy and inform other players about it. First mover advantage states that a player who can become a leader will not be worse off than an original game where all players act simultaneously. Also known as Stackelberg leadership If a player has the power to commit he should do so.

11. First mover advantage TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 “If a player has the power to commit he should do so.” CAUTION: what happens in the case where more than one players can commit to a move? It is not necessarily best go first… What if player 2 was to be leader? “If a player has the power to commit he should do so.” CAUTION: what happens in the case where more than one players can commit to a move? It is not necessarily best go first… What if player 2 was to be leader? Customer VoIP Provider BuyDon’t buy High 2222 1010 Low 0303 1111

12. Chip Duopoly TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Solution by dominance gives (M, m) with a utility of 16 for each player. II I hmln H 0000 8 12 9 18 0 36 M 12 8 16 15 20 0 32 L 18 9 20 15 18 0 27 N 36 0 32 0 27 0 0000

13. Chip Duopoly with commitment TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 0, 0 12, 8 18, 9 36, 0 8, 12 16, 16 20, 15 32, 0 9, 18 15, 20 18,18 27, 0 0, 36 0, 32 0, 27 0, 0 H M L N h m l n Player I has an incentive to commit to strategy H, gaining a utility of 18 and leaving II with 9.

14. Imperfect information TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Players do not always have full access to all information that is relevant to the choices they make. Situations like this are modeled by extensive games with imperfect information. Eg. Small startup announces the development of a key technological product. A big and well established company that dominates the market is known to have a large R&D dept… But others don’t know what the R&D has come up with… The large company can announce the future release of a competitive product or cede the market. The first is a real or bluff option, where the second will not be in the case the product is within reach. The small company after an announcement can chose either stay in or sell out Players do not always have full access to all information that is relevant to the choices they make. Situations like this are modeled by extensive games with imperfect information. Eg. Small startup announces the development of a key technological product. A big and well established company that dominates the market is known to have a large R&D dept… But others don’t know what the R&D has come up with… The large company can announce the future release of a competitive product or cede the market. The first is a real or bluff option, where the second will not be in the case the product is within reach. The small company after an announcement can chose either stay in or sell out

15. Imperfect information TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Chance ( 20, -4 ) ( 12, 4 ) ( -4, 20 ) ( 12, 4 ) ( 0, 16 ) x% strong (100-x)% weak announce cede stay in sell out stay in sell out ( 12, 4 )

16. Imperfect information TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Backward induction can no longer work so: Games with imperfect information do not have an equilibrium in pure strategies… Randomization is in order and the way to do it is by taking the strategic form of the game with expected payoffs Backward induction can no longer work so: Games with imperfect information do not have an equilibrium in pure strategies… Randomization is in order and the way to do it is by taking the strategic form of the game with expected payoffs Players can not distinguish among discrete nodes in an information set. He must make the same move at each node in an information set. Players can not distinguish among discrete nodes in an information set. He must make the same move at each node in an information set. ( 20, -4 ) ( 12, 4 ) ( -4, 20 ) ( 12, 4 ) ( 0, 16 ) x% strong (100-x)% weak announce cede stay in sell out stay in sell out ( 12, 4 )

17. Zero sum games TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 The case of players playing fully opposed interest games embodies the class of two-player zero sum games. (rock-paper-scissors, chess, poker, pistol duels etc) In zero sum games the sum of all player utilities at each game solution is 0. An over-class are constant-sum games, where the sum of all utilities in each game solution is constant. Mixed strategies are intuitively the best solution for constant sum games with imperfect information. Usage of zero sum games along with randomized algorithms in online computation analysis. The case of players playing fully opposed interest games embodies the class of two-player zero sum games. (rock-paper-scissors, chess, poker, pistol duels etc) In zero sum games the sum of all player utilities at each game solution is 0. An over-class are constant-sum games, where the sum of all utilities in each game solution is constant. Mixed strategies are intuitively the best solution for constant sum games with imperfect information. Usage of zero sum games along with randomized algorithms in online computation analysis.

18. Auctions & bidding TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 English (open ascending bid) auction: “ Do I have fifty?…Yes!…Do I have fifty-five? ” Simplified version: Second price auction offers made once over secure channels Highest bidder wins and pays the value of the second bidder. How to bid in a second price auction ? Use your private value of the object English (open ascending bid) auction: “ Do I have fifty?…Yes!…Do I have fifty-five? ” Simplified version: Second price auction offers made once over secure channels Highest bidder wins and pays the value of the second bidder. How to bid in a second price auction ? Use your private value of the object

19. Bidding private value TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Private value for an object is derived from needs and potential resell value estimated by each bidder. Bidding the private value in a second-price auction is a weakly dominant strategy. ie: regardless how others bid, no other strategy will get the bidder of private value better off. Examine what happens in different strategies: Bid lower than your private price Bid higher than your private price Same rationale works for the English auction Private value for an object is derived from needs and potential resell value estimated by each bidder. Bidding the private value in a second-price auction is a weakly dominant strategy. ie: regardless how others bid, no other strategy will get the bidder of private value better off. Examine what happens in different strategies: Bid lower than your private price Bid higher than your private price Same rationale works for the English auction

20. Common values TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Private values is most auctions a non realistic assumption. Art objects / memorabilia may be bought as investments but radio spectrum license is bought for business. The license value depends on market forces such as demand for mobile telephony. Such forces have a more or less common impact to all bidders. So we say that such auctions have common value aspects. What happens when we have pure common value? Private values is most auctions a non realistic assumption. Art objects / memorabilia may be bought as investments but radio spectrum license is bought for business. The license value depends on market forces such as demand for mobile telephony. Such forces have a more or less common impact to all bidders. So we say that such auctions have common value aspects. What happens when we have pure common value?

21. The Winner’s Curse TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 Uncertainty about the value jumps in. For our spectrum auction each company will conducted its own private market research to estimate retail demand and each will come with slightly different results… The results of these estimates is correct on average, but the highest (winning) estimate and the lowest will deviate from the real value.. So playing like this and winning is bad news: the value has been overestimated and this is the winner’s curse… Uncertainty about the value jumps in. For our spectrum auction each company will conducted its own private market research to estimate retail demand and each will come with slightly different results… The results of these estimates is correct on average, but the highest (winning) estimate and the lowest will deviate from the real value.. So playing like this and winning is bad news: the value has been overestimated and this is the winner’s curse…

22. Further Reading TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003 www.gametheory.net R. Gibbons, Game Theory for Applied Economists Osborne, An Introduction to Game Theory Osborn and Rubinstein, A Course in Game Theory R. Axelrod, The Evolution of Cooperation Θεωρία Παιγνίων στο Ρέθυμνο http://www.soc.uoc.gr/petrakis/8ewria.php Marcus M. Mobius’ Game Theory Course http://www.courses.fas.harvard.edu/~ec1052/ The Role of Game Theory in Ad hoc Networks http://www.cs.ucsb.edu/~ebelding/courses/595/s04_gametheory/ Juha Leino Applications of Game Theory in Ad Hoc Networks www.gametheory.net R. Gibbons, Game Theory for Applied Economists Osborne, An Introduction to Game Theory Osborn and Rubinstein, A Course in Game Theory R. Axelrod, The Evolution of Cooperation Θεωρία Παιγνίων στο Ρέθυμνο http://www.soc.uoc.gr/petrakis/8ewria.php Marcus M. Mobius’ Game Theory Course http://www.courses.fas.harvard.edu/~ec1052/ The Role of Game Theory in Ad hoc Networks http://www.cs.ucsb.edu/~ebelding/courses/595/s04_gametheory/ Juha Leino Applications of Game Theory in Ad Hoc Networks

23. TNL - Mobile Computing Group Angelakis Vangelis 21/10/2003