Dominant and Dominated Strategies

Dominant and Dominated Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Junel 8th, 2016 C. Hurtado (UIUC - Economics) Game Theory

On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

The Extensive Form Representation of a Game On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

The Extensive Form Representation of a Game What is a Game? From the noncooperative point of view, a game is a multi-person decision situation defined by its structure, which includes: - Players: Independent decision makers - Rules: Which specify the order of players decisions, their feasible decisions at each point they are called upon to make one, and the information they have at such points. - Outcome: How players decisions jointly determine the physical outcome. - Preferences: players preferences over outcomes. C. Hurtado (UIUC - Economics) Game Theory 1 / 39

The Extensive Form Representation of a Game Examples Matching Pennies (version A). Players: There are two players, denoted 1 and 2. Rules: Each player simultaneously puts a penny down, either heads up or tails up. Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise, player 2 pays 1 dollar to player 1. Matching Pennies (version B). Players: There are two players, denoted 1 and 2. Rules: Player 1 puts a penny down, either heads up or tails up. Then, Player 2 puts a penny down, either heads up or tails up. Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise, player 2 pays 1 dollar to player 1. C. Hurtado (UIUC - Economics) Game Theory 2 / 39

The Extensive Form Representation of a Game Examples Matching Pennies (version C). Players: There are two players, denoted 1 and 2. Rules: Player 1 puts a penny down, either heads up or tails up, without letting player 2 know his decision. Player 2 puts a penny down, either heads up or tails up. Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise, player 2 pays 1 dollar to player 1. Matching Pennies (version D). Players: There are two players, denoted 1 and 2. Rules: Players flip a fair coin to decide who begins. The looser puts a penny down, either heads up or tails up. Then, the winner puts a penny down, either heads up or tails up. Outcomes: If the two pennies match, the looser pays 1 dollar to player 2; otherwise, the winner pays 1 dollar to player 1. C. Hurtado (UIUC - Economics) Game Theory 3 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game Some games that are important in economics have simultaneous moves. Simultaneous means strategically simultaneous, in the sense that players decisions are made without knowledge of others decisions. It need not mean literal synchronicity, although that is sufficient for strategic simultaneity. But many important games have at least some sequential decisions, with some later decisions made with knowledge of others earlier decisions. We need a way to describe and analyze both kinds of game. One way to describe either kind of game is via the extensive form or game tree, which shows a game s sequence of decisions, information, outcomes, and payoffs. C. Hurtado (UIUC - Economics) Game Theory 4 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game A version of Matching Pennies with sequential decisions, in which Player 1 moves first and player 2 observes 1 s decision before 2 chooses his decision. C. Hurtado (UIUC - Economics) Game Theory 5 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game We can represent the usual Matching Pennies with simultaneous decisions by introducing an information set, which includes the decision nodes a player cannot distinguish and at which he must therefore make the same decision, as in the circled nodes. C. Hurtado (UIUC - Economics) Game Theory 6 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game The order in which simultaneous decision nodes are listed has some flexibility, as in previous case, where player 2 could have been at the top. For sequential decisions the order must respect the timing of information flows. (Information about decisions already made, as opposed to predictions of future decisions, has no reverse gear.) All decision nodes in an information set must belong to the same player and have the same set of feasible decisions. (Why?) Players are normally assumed necessarily to have perfect recall of their own past decisions (and other information). If so, the tree must reflect this. Definition A game is one of perfect information if each information set contains a single decision node. Otherwhise, it is a game of imperfect information. C. Hurtado (UIUC - Economics) Game Theory 7 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game This is an example of a game with simultaneous decision nodes and players with perfect recall of their own past decisions. C. Hurtado (UIUC - Economics) Game Theory 8 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game This is an example of a game with simultaneous decision nodes and players without perfect recall of their own past decisions. C. Hurtado (UIUC - Economics) Game Theory 9 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game This is another example of a game with simultaneous decision nodes and players without perfect recall of their own past decisions. C. Hurtado (UIUC - Economics) Game Theory 10 / 39

The Extensive Form Representation of a Game The Extensive Form Representation of a Game Shared uncertainty (in economics symmetric information ) can be modeled by introducing moves by an artificial player (without preferences) called Nature, who chooses the structure of the game randomly, with commonly known probabilities. C. Hurtado (UIUC - Economics) Game Theory 11 / 39

Strategies and the Normal Form Representation of a Game On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game For sequential games it is important to distinguish strategies from decisions or actions. A strategy is a complete contingent plan for playing the game, which specifies a feasible decision for each of a player s information sets in the game. Recall that his decision must be the same for each decision node in an information set. A strategy is like a detailed manual of actions, not like a single decision or action. C. Hurtado (UIUC - Economics) Game Theory 12 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game C. Hurtado (UIUC - Economics) Game Theory 13 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game It is assumed that conditional on what a player observes, he can predict the probability distributions of his own and others future decisions and their consequences. If players have this kind of foresight, then their rational sequential decision-making in real time should yield exactly the same distribution of decisions as simultaneous choice of fully contingent strategies at the start of play. The player writes his own manual of actions. Then he will give you (a neutral referee) the manual and let you play out the game. You will tell him who won. Because strategies are complete contingent plans, players must be thought of as choosing them simultaneously (without observing others strategies), independently, and irrevocably at the start of play. C. Hurtado (UIUC - Economics) Game Theory 14 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game Why a strategy must be a complete contingent plan, specifying decisions even for a player s own nodes that he knows will be ruled out by his own earlier decisions? Otherwise, other players strategies would not contain enough information for a player to evaluate the consequences of his own alternative strategies. We would then be unable to correctly formalize the idea that a strategy choice is rational. Putting the point in an only seemingly different way, in individual decision theory, zero probability events can be ignored as irrelevant, at least for expected-utility maximizers. But in games zero-probability events cannot be ignored because what has zero probability is endogenously determined by players strategies. C. Hurtado (UIUC - Economics) Game Theory 15 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game Player 2 strategies: Strategy 1 (s 1): Play H if player 1 plays H; Play H if player 1 plays T Strategy 2 (s 2): Play H if player 1 plays H; Play T if player 1 plays T Strategy 3 (s 3): Play T if player 1 plays H; Play H if player 1 plays T Strategy 4 (s 4): Play T if player 1 plays H; Play T if player 1 plays T C. Hurtado (UIUC - Economics) Game Theory 16 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game A game maps strategy profiles (one for each player) into payoffs (with outcomes implicit). A game form maps strategy profiles into outcomes, without specifying payoffs. Specifying strategies make it possible to describe an extensive-form game s relationship between strategy profiles and payoffs by its (unique) normal form or payoff matrix or (usually when strategies are continuously variable) payoff function. C. Hurtado (UIUC - Economics) Game Theory 17 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game C. Hurtado (UIUC - Economics) Game Theory 18 / 39

Strategies and the Normal Form Representation of a Game Strategies and the Normal Form Representation of a Game The mapping from the normal to the extensive form isn t univalent: the normal form for Matching Pennies version B has possible extensive forms other than the one depicted before: C. Hurtado (UIUC - Economics) Game Theory 19 / 39

Randomized Choices On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Randomized Choices Randomized Choices In game theory it is useful to extend the idea of strategy from the unrandomized (pure) notion we have considered to allow mixed strategies (randomized strategy choices). Example: Matching Pennies Version A has no appealing pure strategies, but there is a convincingly appealing way to play using mixed strategies: randomizing 50-50. (Why?) Our definitions apply to mixed as well as pure strategies, given that the uncertainty about outcomes that mixed strategies cause is handled (just as for other kinds of uncertainty) by assigning payoffs to outcomes so that rational players maximize their expected payoffs. Mixed strategies will enable us to show that (reasonably well-behaved) games always have rational strategy combinations. In extensive-form games with perfect recall, mixed strategies are equivalent to behavior strategies, probability distributions over pure decisions at each node (Kuhn s Theorem; see MWG problem 7.E.1). C. Hurtado (UIUC - Economics) Game Theory 20 / 39

Exercises On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Exercises Exercises Exercise 1. In a game where player i has N information sets indexed n = 1,, N and M n possible actions at information set n, how many strategies does player i have? Exercise 2. Depict the normal formm of Matching Pennies Version C. C. Hurtado (UIUC - Economics) Game Theory 21 / 39

Exercises Exercises Exercise 3. Consider the followign two-player (excluding payoffs): a) What are player 1 s possible strategies? player 2 s? b) Suppose that we change the game by merging the information set of player 1 s second round of moves (so that all the four nodes are now in a single information set). Argue why the game is no longer one of perfect recall. C. Hurtado (UIUC - Economics) Game Theory 22 / 39

Formalizing the Game On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Formalizing the Game Formalizing the Game Up to this point we defined game without been formal. Let me introduce some Notation: - set of players: I = {1, 2,, N} - set of actions: i I, a i A i, where each player i has a set of actions A i. - strategies for each player: i I, s i S i, where each player i has a set of pure strategies S i available to him. A strategy is a complete contingent plan for playing the game, which specifies a feasible action of a player s information sets in the game. - profile of pure strategies: s = (s 1, s 2,, s N ) N i=1 Si. Note: let s i = (s 1, s 2,, s i 1, s i+1,, s N ) S i, we will denote s = (s i, s i) (S i, S i). - Payoff function: u i : N Si R, denoted by ui(si, s i) i=1 C. Hurtado (UIUC - Economics) Game Theory 23 / 39

Formalizing the Game Formalizing the Game Now we can denote game with pure strategies and complete information in normal form by: Γ N = {I, {S i} i, {u i} i}. What about the games with mix strategies? We have taken it that when a player acts at any information set, he deterministically picks an action from the set of available actions. But there is no fundamental reason why this has to be case. Definition A mixed strategy for player i is a function σ i : S i [0, 1], which assigns a probability σ i(s i) 0 to each pure strategy s i S i, satisfying s i S i σ i(s i) = 1. We denote the set of mixed strategies by (S i). Note that a pure strategy can be viewed as a special case of a mixed strategy in which the probability distribution over the elements of S i is degenerate. C. Hurtado (UIUC - Economics) Game Theory 24 / 39

Formalizing the Game Example Meeting in New York: - Players: Two players, 1 and 2 - Rules: The two players can not communicate. They are suppose to meet in NYC at noon to have lunch but they have not specify where. Each must decide where to go (only one choice). - Outcomes: If they meet each other, they enjoy other s company. Otherwise, they eat alone. - Payoffs: They attach a monetary value of 100 USD to other s company and 0 USD to eat alone. player 1 player 2 A B C A 100,100 0,0 0,0 B 0,0 100,100 0,0 C. Hurtado (UIUC - Economics) Game Theory 25 / 39

Formalizing the Game Example Meeting in New York: - set of players: I = {1, 2} - set of actions: A 1 = {A, B}, and A 2 = {A, B, C} - strategies for each player: S 1 = A 1, and S 2 = A 2 (Why?) - Payoff function: u i : 2 Si R, denoted by ui(si, s i) i=1 u(s i, s i) = { 100 0 if s i = s i if s i s i Player 2 - pure strategies: S 2 = {A, B, C}. Player 2 has 3 pure strategies. - mixed strategies: (S 2) = {(σ 2 1, σ 2 2, σ 2 3) R 3 σ 2 m 0 m = 1, 2, 3 and 3 m=1 σ2 m = 1} C. Hurtado (UIUC - Economics) Game Theory 26 / 39

Dominant and Dominated Strategies On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Dominant and Dominated Strategies Dominant and Dominated Strategies Now we turn to the central question of game theory: What should be expected to observe in a game played by rational agents who are fully knowledgeable about the structure of the game and each others rationality? To keep matters simple we initially ignore the possibility that players might randomize in their strategy choices. The prisoner s dilemma: * Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. * The prosecutors do not have enough evidence to convict the pair on the principal charge. They hope to get both sentenced to a year in prison on a lesser charge. * Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. * Here is the offer: - If A and B each betray the other, each of them serves 2 years in prison - If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vice versa) - If A and B both remain silent, both of them will only serve 1 year in prison (on the lesser charge) C. Hurtado (UIUC - Economics) Game Theory 27 / 39

Dominant and Dominated Strategies Dominant and Dominated Strategies Let me put prisoner s dilemma as a game of trust: player 1 player 2 trust cheat trust 5,5 1,10 cheat 10,1 2,2 Observe that regardless of what her opponent does, player i is strictly better off playing Cheat rather than Trust. This is precisely what is meant by a strictly dominant strategy. Player 2 plays Trust. Player 1 knows that 10 > 5, better to Cheat. Player 2 plays Cheat. Player 1 knows that 2 > 1, better to Cheat. Regardless of the other s strategies, it is always better to Cheat. Note that both would be better off if they both play trust. Lesson: self-interested behavior in games may not lead to socially optimal outcomes. C. Hurtado (UIUC - Economics) Game Theory 28 / 39

Dominant and Dominated Strategies Dominant and Dominated Strategies Definition A strategy s i S i is a strictly dominant strategy for player i if for all s i s i and all s i S i, u i (s i, s i ) > u i ( s i, s i ). A strictly dominant strategy for i uniquely maximizes her payoff for any strategy profile of all other players. If such a strategy exists, it is highly reasonable to expect a player to play it. In a sense, this is a consequence of a player s rationality. C. Hurtado (UIUC - Economics) Game Theory 29 / 39

Dominant and Dominated Strategies Dominant and Dominated Strategies What about if a strictly dominant strategy doesn t exist? player 1 player 2 a b c A 5,5 0,10 3,4 B 3,0 2,2 4,5 You can easily convince yourself that there are no strictly dominant strategies here for either player. Notice that regardless of whether Player 1 plays A or B, Player 2 does strictly better by playing b rather than a. That is, a is strictly dominated by b. C. Hurtado (UIUC - Economics) Game Theory 30 / 39

Dominant and Dominated Strategies Dominant and Dominated Strategies Definition A strategy s i S i is strictly dominated for player i if there exists a strategy s i S i such that for all s i S i, u i ( s i, s i ) > u i (s i, s i ). In this case, we say that s i strictly dominates s i. In words, s i strictly dominates s i if it yields a strictly higher payoff regardless of what (pure) strategy rivals use. Note that the definition would also permits us to use mixed strategies Using this terminology, we can restate the definition of strictly dominant: A strategy s i is strictly dominant if it strictly dominates all other strategies. It is reasonable that a player will not play a strictly dominated strategy, a consequence of rationality, again. C. Hurtado (UIUC - Economics) Game Theory 31 / 39

Iterated Delation of Strictly Dominated Strategies On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Iterated Delation of Strictly Dominated Strategies Iterated Delation of Strictly Dominated Strategies player 1 player 2 a b c A 5,5 0,10 3,4 B 3,0 2,2 4,5 We argued that a is strictly dominated (by b) for Player 2; hence rationality of Player 2 dictates she won t play it. We can push the logic further: if Player 1 knows that Player 2 is rational, he should realize that Player 2 will not play strategy a. Notice that we are now moving from the rationality of each player to the mutual knowledge of each player s rationality. Once Player 1 realizes that 2 will not play a and deletes this strategy from the strategy space, then strategy A becomes strictly dominated by strategy B for Player 2. If we iterate the knowledge of rationality once again, then Player 2 realizes that 1 will not play A, and hence deletes A. Player 2 should play c. We have arrived at a solution. C. Hurtado (UIUC - Economics) Game Theory 32 / 39

Iterated Delation of Strictly Dominated Strategies Iterated Delation of Strictly Dominated Strategies Definition A game is strict-dominance solvable if iterated deletion of strictly dominated strategies results in a unique strategy profile. Since in principle we might have to iterate numerous times in order to solve a strict-dominance solvable game, the process can effectively can only be justified by common knowledge of rationality. As with strictly dominant strategies, it is also true that most games are not strict-dominance solvable. You might worry whether the order in which we delete strategies iteratively matters. Insofar as we are working with strictly dominated strategies so far, it does not. C. Hurtado (UIUC - Economics) Game Theory 33 / 39

Iterated Delation of Dominated Strategies On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Iterated Delation of Dominated Strategies Iterated Delation of Dominated Strategies Definition A strategy s i S i is a weakly dominant strategy for player i if for all s i s i and all s i S i, u i (s i, s i ) u i ( s i, s i ), and for at least one choice of s i the inequality is strict. Definition A strategy s i S i is weakly dominated for player i if there exists a strategy s i S i such that for all s i S i, u i ( s i, s i ) u i (s i, s i ), and for at least one choice of s i the inequality is strict. In this case, we say that s i weakly dominates s i. Definition A game is weakly-dominance solvable if iterated deletion of weakly dominated strategies results in a unique strategy profile. C. Hurtado (UIUC - Economics) Game Theory 34 / 39

Iterated Delation of Dominated Strategies Iterated Delation of Dominated Strategies Using this terminology, we can restate the definition of weakly dominant: A strategy s i is weakly dominant if it weakly dominates all other strategies. You might worry whether the order in which we delete strategies iteratively matters. Delation of dominated strategies could leave to different outcomes. P2 L R U 5,1 4,0 P1 M 6,0 3,1 D 6,4 4,4 P2 P2 L R L R P1 U 5,1 4,0 D 6,4 4,4 P1 M 6,0 3,1 D 6,4 4,4 C. Hurtado (UIUC - Economics) Game Theory 35 / 39

Exercises On the Agenda 1 The Extensive Form Representation of a Game 2 Strategies and the Normal Form Representation of a Game 3 Randomized Choices 4 Exercises 5 Formalizing the Game 6 Dominant and Dominated Strategies 7 Iterated Delation of Strictly Dominated Strategies 8 Iterated Delation of Dominated Strategies 9 Exercises C. Hurtado (UIUC - Economics) Game Theory

Exercises Exercises Exercise 1. Prove that a player can have at most one strictly dominant strategy. Exercise 2. Apply the iterated elimination of strictly dominated strategies to the following normal form games. Note that in some cases there may remain more that one strategy for each player. Say exactly in what order you eliminated rows and columns. Exercise 3. Apply the iterated elimination of dominated strategies to the following normal form games. Note that in some cases there may remain more that one strategy for each player. Say exactly in what order you eliminated rows and columns. C. Hurtado (UIUC - Economics) Game Theory 36 / 39

Exercises Exercises Exercise 2 (cont.). C. Hurtado (UIUC - Economics) Game Theory 37 / 39

Exercises Exercises Exercise 2 (cont.). C. Hurtado (UIUC - Economics) Game Theory 38 / 39

Exercises Exercises Exercise 2 (cont.). C. Hurtado (UIUC - Economics) Game Theory 39 / 39