Chapter 15- Game Theory: The Mathematics of Competition



Name:___________________________________________________Date:________ Discrete Math 2: 15.1 Game Theory; Two-person Total Conflict Games (Page 1)Objective(s): The learner will be able to calculate the maximin and minimax strategies from a two-person total-conflict game. In addition, the student will determine if a saddlepoint arises from the strategies.__________________________________________________________________________________________Chapter 15- Game Theory: The Mathematics of CompetitionChapter 15 OutlineTwo-Person Total Conflict Games: Pure StrategiesTwo-Person Total Conflict Games: Mixed StrategiesPartial Conflict GamesLarger GamesExtensive Form Games: A sequence of movesWeb site for further exploration: DefinitionsGame Theory: use of mathematical tools to study situations (games) involving conflict and cooperation.Strategies: courses of action a player can choose in a gameOutcomes: consequences of choice made.Rational Choice: a choice that leads to a preferred outcomeTotal Conflict: (a zero-sum or constant-sum game) when one player wins, the other player losesIntroduction- Fundamental Principles of Game Theory When analyzing any game, we make the following assumptions about both players: ???????? Each player makes the best possible move. ???????? Each player knows that his or her opponent is also making the best possible move.A) Two-person Total Conflict Game: Pure StrategiesA player uses a pure strategy if he or she uses the same strategy at each round of the game. Zero-sum games: Games of “total conflict” – one player’s _________ equals the other player’s ________. If we sum the payoffs at each outcome, the result is always _________.Two-player simultaneous move games (both zero and variable sum types) can be written in ___________ form (also called __________________ form) Example: (p470 picture)Let’s say 2 people (Sally and Bob) are trying to decide on the location of a restaurant near an intersection.Bob wants a higher elevation; Sally wants a lower elevation. They both want an intersection. Sally will choose from the 3 the N-S highways; Bob will choose the 3 E-W highways. The numbers in the chart represent altitude in thousands of feet.SallyBobA higher number is better for Bob and therefore worse for Sally. (Note- In game theory notation, the payoff matrix in a zero-sum game will always be in terms of the row player) Sally will choose 1, 2, or 3 and Bob will choose A, B, or C. Bob wants to get the highest altitude, no matter what route Sally chooses. Sally wants the lowest altitude, no matter what Bob chooses. Which row do you think Bob should select? _________ Which column do you think Sally should select? _________ Now let’s look at the strategy:If Bob selects A, Sally will choose _______ , and the value would be _______ . If Bob selects B, Sally will choose _______ , and the value would be _______ .If Bob selects C, Sally will choose _______ , and the value would be _______ .If Sally selects 1, Bob will choose _______ , and the value would be _______ .If Sally selects 2, Bob will choose _______ , and the value would be _______ .If Sally selects 3, Bob will choose _______ , and the value would be _______ .Based on the strategies, which row should Bob select? _________Based on the strategies, which column should Sally select? _________With rows (which are to be maximized), we use the biggest minimum (maximin).With columns (which are to minimized), we use the smallest maximum (minimax).Saddlepoint - when the maximin and minimax strategies produce an equal valueChoosing the row and column through any saddle point gives ______________ strategies for both players.Examples: Determine if there is a saddlepoint. If so, what is the best strategy?1)Player 2123A826Player 1 B347C554Which row should player 1 pick?Which column should player 2 pick?Is there a saddlepoint? What is it?Player 21234A10172015Player 1 B15151525C20202020D151520202)Which row should player 1 pick?Which column should player 2 pick?Is there a saddlepoint? What is it?Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 2a)For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.1)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A403530B253020Which column should player 2 pick?C252530D202015Is there a saddlepoint? What is it?2)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A100150300200B250220250150Which column should player 2 pick?Is there a saddlepoint? What is it?3)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A30704080B40201070Which column should player 2 pick?C20607040D60503020Is there a saddlepoint? What is it?4)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A67118B25412Which column should player 2 pick?C116315D9141310Is there a saddlepoint? What is it?Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 2b)For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.5)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A342B146Which column should player 2 pick?C978Is there a saddlepoint? What is it?6)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A1573B6245Which column should player 2 pick?C3467Is there a saddlepoint? What is it?7)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A1510164B109817Which column should player 2 pick?C681412D14111513Is there a saddlepoint? What is it?8)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A432B346Which column should player 2 pick?C958Is there a saddlepoint? What is it?Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 3a)For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.1)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A876B564Which column should player 2 pick?C556D443Is there a saddlepoint? What is it?2)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A15181913B18172012Which column should player 2 pick?Is there a saddlepoint? What is it?3)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A79104B8364Which column should player 2 pick?C2443D10685Is there a saddlepoint? What is it?4)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A8956B7654Which column should player 2 pick?C91037D6445Is there a saddlepoint? What is it?Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 3b)For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.5)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A564B673Which column should player 2 pick?C243Is there a saddlepoint? What is it?6)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A10507030B60204050Which column should player 2 pick?C30406070Is there a saddlepoint? What is it?7)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A4161015B178910Which column should player 2 pick?C121486D13151114Is there a saddlepoint? What is it?8)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A203040B604030Which column should player 2 pick?C805090Is there a saddlepoint? What is it?Name:___________________________________________________Date:________ Discrete Math 2: 15.2 Two person Total Conflict Games, Mixed Strategies (Page 1a)Objective(s): The learner will be able to calculate the optimal probabilities for each strategy for each player in a two-person total-conflict game.__________________________________________________________________________________________B) Two-person total conflict game: Mixed strategies (NO saddlepoint!!)A mixed strategy is a method of playing a game where the rows or columns are played at ___________ so that each is used a given fraction of the time. Pure strategies are the actual strategies that players have available to choose from when playing a matrix game. In a simultaneous move (matrix) game, sometimes players can benefit from randomly choosing one or the other of their pure strategies. A mixed strategy is the decision to play each of the pure strategies with some specific _________________. If pure strategies do not produce a ________________, we proceed as follows: Define variables that represent the __________________ each player will play each available strategy. For each player we find the probabilities that will provide the lowest expected payoff for the other player. Example:Player 2HeadsTails Player 1Heads5-2Tails-31 What is Player’s 1 best strategy?What is Player’s 2 best strategy?Is there a saddle point? What does this mean?1) Let’s look at Player 1 first. What is the best that Player 1 can do, regardless of player 2’s choice?Player 1’s expected payoff is:So Player 1 chooses Heads _____ of the time and Tails ____ of the time. The expected payoff for Player 1 is2) And Player 2’s expected payoff is:Player 2 will choose Heads ____ of the time and Tails ______ of the time. The expected payoff for Player 2 is This was a zero-sum game: one in which the payoff to one player is the negative of the corresponding payoff to the other, so the sum of the payoffs is always zero.Example: When a pitcher and batter face each other in a baseball game, we can consider each pitch as a simultaneous move zero-sum game. In this situation, the pitcher can throw a fastball or a curve ball. The batter will swing after guessing that the pitcher will throw a fastball or a curve ball. The numbers shown represent batting averages. Find the optimal mixed strategy for the pitcher and for the batter.?FastballCurve BatterFastball.300.200Curve.100.500PitcherWhich row should the Batter pick?Which column should Pitcher pick?Is there a saddlepoint? What does this mean? What % should Batter choose Fastball? What % should Batter choose Curveball?What % should Pitcher choose Fastball? What % should Pitcher choose Curveball?__________________________________________________________________________________________Example:(Player 2)(Player 1)ColumnsRows12 Which row should player 1 pick?A4760 Which column should player 1 pick?B2831 Is there a saddlepoint? What does this mean?Find the mixed strategy for each player: (Player 2)What % should P1 choose row A?(Player 1)ColumnsWhat % should P1 choose row B?Rows12What % should P2 choose column 1?A4760qWhat % should P2 choose column 2?B2831(1 – q)p(1 – p)Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.2– Two-Person Zero-Sum Games, Mixed Strategies (Page 2a)For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.1)(Player 2)(Player 1)ColumnsRows123Which row should player 1 pick?A181716B151614Which column should player 2 pick?C151516D141413Is there a saddlepoint? What is it?2)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A30363826B36344024Which column should player 2 pick?Is there a saddlepoint? What is it?3)(Player 2)(Player 1)ColumnsRows1234Which row should player 1 pick?A5446B4567Which column should player 2 pick?C73109D8925Is there a saddlepoint? What is it?4) (Player 2)(Player 1) ColumnsRows12 Which row should player 1 pick?A350450 Which column should player 2 pick?B400150 Is there a saddlepoint? What does this mean? What % should P1 choose row A?What % should P1 choose row B?What % should P2 choose column 1?What % should P2 choose column 2?Discrete Math 2: 15.2– Two-Person Zero-Sum Games, Mixed Strategies (Page 2b)For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.5) (Player 2)(Player 1)ColumnsRows12Which row should player 1 pick?A68Which column should player 1 pick?B102 Is there a saddlepoint? What does this mean? What % should P1 choose row A? What % should P1 choose row B?What % should P2 choose column 1? What % should P2 choose col 2?6) ?FastballCurve BatterFastball.400.200Curve.100.500PitcherWhich row should the Batter pick?Which column should Pitcher pick?Is there a saddlepoint? What % should Batter choose Fastball? What % should Batter choose Curveball?What % should Pitcher choose Fastball? What % should Pitcher choose Curveball?7) When it is third down and short yardage to go for a first down in American football, the quarterback can decide to run the ball or pass it. Similarly the other team can prepare a heavier defense against a run or pass. The following matrix displays the probabilities of obtaining a first down. Defense?RunPassOffenseRun0.50.8Pass0.70.2Which row should the Offense pick?Which column should Defense pick?Is there a saddlepoint? What % should Offense choose to Run? What % should Offense choose to Pass?What % should Defense choose Run? What % should Defense choose Pass?Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 3a)Find the optimal mixed strategy for each participant:1) Player 1 is trying to maximize and Player 2 is trying to minimize. Player 212Player 1A510qB76(1 – q)p(1 – p)2) PitcherFCBatterF.250.150qC.200.400(1 – q)p(1 – p)Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 3b)3) It’s a soccer penalty kick! Should the shooter kick to the side or the center and which way should the goalie guess? The number represents the probability of the kick being made. Specify the best strategies for each player.GoalieSideCenterShooterSide.4.9qCenter.8.3(1 – q)p(1 – p)4) Two thieves were caught. They must to decide whether to confess or to keep quiet. The numbers listed show the length of sentence each would receive: Thief 2QuietConfessThief 1Quiet06qConfess63(1 – q)p(1 – p)Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 4a)NEW SHORT-CUT:Assume the values are in the grid as:ABCDp = q = 1) Player 1 is trying to maximize and Player 2 is trying to minimize. Player 212Player 1A83qB49(1 – q)p(1 – p)What fraction/percent should:Player 1 choose A? __________Player 1 choose B? __________Player 2 choose 1? __________Player 2 choose 2? __________2) PitcherFCBatterF.400.200qC.100.500(1 – q)p(1 – p)What fraction/percent should:Batter guess Fastball? __________Batter guess Curve? __________Pitcher throw Fastball? __________Pitcher throw Curve? __________Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 4b)3) It’s a soccer penalty kick! Should the shooter kick to the side or the center and which way should the goalie guess? The number represents the probability of the kick being made. Specify the best strategies for each player.GoalieSideCenterShooterSide50%80%qCenter90%20%(1 – q)p(1 – p)What fraction/percent should:Goalie guess Side? __________Goalie guess Center? __________Shooter kick Side? __________Shooter kick Center? __________4) Two thieves were caught. They must to decide whether to confess or to keep quiet. The numbers listed show the length of sentence each would receive: Thief 2QuietConfessThief 1Quiet515qConfess1510(1 – q)p(1 – p) Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 5a)1) Player 1 is trying to maximize and Player 2 is trying to minimize. Player 212Player 1A73qB29(1 – q)p(1 – p)What fraction/percent should:Player 1 choose A? __________Player 1 choose B? __________Player 2 choose 1? __________Player 2 choose 2? __________2) PitcherFCBatterF.400.300qC.200.500(1 – q)p(1 – p)What fraction/percent should:Batter guess Fastball? __________Batter guess Curve? __________Pitcher throw Fastball? __________Pitcher throw Curve? __________Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 5b)3) It’s a soccer penalty kick! Should the shooter kick to the side or the center and which way should the goalie guess? The number represents the probability of the kick being made. Specify the best strategies for each player.GoalieSideCenterShooterSide40%90%qCenter80%30%(1 – q)p(1 – p)What fraction/percent should:Goalie guess Side? __________Goalie guess Center? __________Shooter kick Side? __________Shooter kick Center? __________4) Two thieves were caught. They must to decide whether to confess or to keep quiet. The numbers listed show the length of sentence each would receive: Thief 2QuietConfessThief 1Quiet614qConfess1510(1 – q)p(1 – p) Name:___________________________________________________Date:_______Discrete Math 2: 15.3 Two person Partial Conflict Games, Mixed Strategies (Page 1)Objective(s): The learner will be able to calculate the Dominant Strategy for each player in a two-person partial-conflict game.__________________________________________________________________________________________C) Partial Conflict GamesIn the real world, not every player’s loss is another player’s gain. For example, both may benefit or both may lose. Games of partial conflict are variable-sum games, in which the sum of payoffs to the players at the different outcomes varies.Team AwesomeStrategy AStrategy B Team All-StarStrategy 1(2,2)(4,1)Strategy 2(1,4)(3,-3)Partial Conflict: a variable-sum game in which both players can benefit by cooperation but may have strong incentives not to cooperateReal-world applications: labor-management disputes, international crises, Arms raceNow, let’s play another game!!Assuming player I and II move simultaneously, what should they pick?If you are player I, what is your strategy to optimize your payoff? Do you pick 1 or 2? If you are player II, what is your strategy? Is it better to pick A or B?In this case, both teams have Dominant Strategies. The dominant strategies are Strategy 1 for All-Star(1 is ALWAYS better than 2) and Strategy A for Awesome (A is ALWAYS better than B).Dominant Strategy- a strategy that is sometimes better and never worse for a player than every other strategy, whatever strategies the other players chooseExample: Does either player have a dominant strategy in the game below?Player 2 Player 21)ABX1,31,1Y1,21,4Z1,01,12)ABX0,13,0Y-5,51,3Z1,46,-3Player 1 Player 1This leads to a special type of game called the Prisoner’s Dilemma.The game got its name from the following hypothetical situation: imagine two criminals arrested under the suspicion of having committed a crime together. However, the police does not have sufficient proof in order to have them convicted. The two prisoners are isolated from each other, and the police visit each of them and offer a deal: the one who offers evidence against the other one will be freed. If none of them accepts the offer, they are in fact cooperating against the police, and both of them will get only a small punishment because of lack of proof. They both gain. However, if one of them betrays the other one, by confessing to the police, the defector will gain more, since he is freed; the one who remained silent, on the other hand, will receive the full punishment, since he did not help the police, and there is sufficient proof. If both betray, both will be punished, but less severely than if they had refused to talk. The dilemma resides in the fact that each prisoner has a choice between only two options, but cannot make a good decision without knowing what the other one will do. Using the matrix above, did either criminal have a dominant strategy?ConfessNot ConfessConfess(2,2)(4,1)Not Confess(1,4)(3,3)Let’s play another game!! Does either player have a dominant strategy?1) This game is an example of a type of game called Chicken.2) Arms race. Lets say we have 2 countries Chicken: a two-person, variable-sum symmetric game in (Red and Blue) that hate each other. They have 2 which each player has two strategies: to swerve to avoid a choices about how to handle their arms. 1) arm ‘collision’ or not to swerve if the other swerves.in prep for possible war 2) disarm or at least try to negotiate arms-control agreement Player 2RedSwerveNot SwerveSwerve(3,3)(2,4) Not Swerve(4,2)(1,1)Continue to ArmRespect TreatyContinue to Arm(2,2)(4,1)Respect Treaty(1,4)(3,3)3) The Cuban missile crisis can be modeled by the game of chicken. In the 1960s, the USSR began supplying missiles to Cuba. The United States began a blockade to stop the USSR. As the crisis developed, if each had continued to proceed with their chosen strategy, the consequences could have been disastrous. Fortunately, some last minute negotiations averted such an outcome. The U.S. did not have to back down and the USSR was able to achieve some advantage in other interests.USSRback downproceedback down(3,3)(2,4)proceed(4,2)(0,0)Name:___________________________________________________Date:________ Discrete Math 2: 15.3 Two person Partial Conflict Games, Mixed Strategies (Page 2a)Objective(s): The learner will be able to calculate the Dominant Strategy for each player in a two-person partial-conflict game.__________________________________________________________________________________________Is there a dominant strategy for either player? If so, what is it?1)Player 2L RU 10,5 20,9Player 1 M 20,10 30,5D 30,10 6,52)Prisoner 2Confess DenyPrisoner 1 Confess -10, -10 -1, -25Deny -25, -1 -3, -33) Player 2X YA 5,2 4,2Player 1B 3,1 3,2C 2,1 4,1D 4,3 5,44)Player 2X YA 6,3 4,3Player 1 B 3,1 3,3C 2,1 4,1D 4,4 5,4Discrete Math 2: 15.3 Two person Partial Conflict Games, Mixed Strategies (Page 2b)5) Player 2L RPlayer 1 M 20,10 30,5D 30,10 5,56) Player 2V W X Y ZA 4,-1 3,0 -3,1 -1,4 -2,0B -1,1 2,2 2,3 -1,0 2,5Player 1 C 2,1 -1,-1 0,4 4,-1 0,2D 1,6 -3,0 -1,4 1,1 -1,4E 0,0 1,4 -3,1 -2,3 -1,-17)Player 2L RPlayer 1 U 5, 5 2, 1D 4, 7 3, 68) Player 2L RPlayer 1 U 3, 1 0, 0D 0, 0 1, 39) P2V W X Y ZA 4,-1 4,2 -3,1 -1,2 -2,0B -1,1 2,2 2,3 -1,0 2,5P1 C 2,3 -1,-1 0,4 4,-1 0,2D 1,3 4,4 -1,4 1,1 -1,2E 0,0 1,4 -3,1 -2,3 -1,-1Name:___________________________________________________Date:________ Discrete Math 2: 15.4 Game Theory; Truels (Page 1)Objective(s): The learner will be able to calculate the Dominant Strategy for each player in a truel game and use trees to determine outcomes of extensive games.__________________________________________________________________________________________D: Extensive Form Games: Game TreesSometimes, the game may not be ‘simultaneous choice’ but instead of sequence of choices (these are called sequential move games and are often modeled using a game tree)The game tree is referred to as the extensive form of the game.Tic tac toe, for example, could be modeled in a game tree. (a very very large one)Example:Let’s say there are 2 monkeys and 1 tree with 5 bananas. There is a Big Monkey and a Little Monkey. One monkey will move first, then the other after.Let’s use Backwards Induction to solve this.What are little monkey’s strategies??What are Big Monkey’s Strategies?What is the resulting payoff?Would this change if we changed the order?A Truel is like a duel, except there are three players. Each player can either fire, or not fire, his gun at either of the other two players. We assume the goal of each player is, first, to survive and, second, to survive with as few other players as possible. Each player has one bullet and is a perfect shot; no communication (for example, to pick out a common target) leading to a binding agreement with other players is allowed, making the game non-cooperative. We will discuss the answers that simultaneous choices, on the one hand, and sequential choices, on the other, give to what is optimal for the players to do in the truel. If choices are simultaneous, at the start of play, each player will fire at one of the other two players, killing that player. Why will the players all fire at each other? What should each player do? This analysis suggests that truels might be more effective than duels in preventing the outbreak of conflict.The game, and optimal strategies in it, would change if the players (1) were allowed more options, such as to fire in the air and thereby disarm themselves, and (2) did not have to choose simultaneously but, instead, a particular order of play were specified. Thus, if the order of play were A, followed by B and C choosing simultaneously. If a players does not shoot the game ends. What should each player do? Let’s create a game tree.Key(x, y, z) = (payoff to A, payoff to B, payoff to C).4 = best, 3 = next best, 2 = next worst, 1 = worst.NS = not shoot; S X = shoot X.Order may lead to radically different outcomes compared with those based on simultaneous choices.In 1992, a modified version of this scenario was played out in late-night television programming among the three major TV broadcasting networks of the time, with ABC’s effectively going first with Nightline, its well-established news program, and CBS’s and NBC’s dueling about which host, David Letterman or Jay Leno, to choose for their entertainment shows. What happened?Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.4 – Game Theory; Truels (Page 2)You are in a truel – a duel involving three people instead of two! In this situation, each player (A, B, C) has a gun with one bullet and never misses whatever target he/she selects. Each person’s goal is to survive. No communication is allowed by any of the players, so no collusion is allowed. What would you do as each player in the following scenarios?1) All players act simultaneously – a shot must be fired and no missing is allowedWhat would you do as:Player A: Player B: Player C: What would be the resulting outcome? (Who would survive?)2) All players act simultaneously – a shot must be fired but missing is allowedWhat would you do as:Player A: Player B: Player C: What would be the resulting outcome? (Who would survive?)3) All players act simultaneously – one can choose not to fire and missing is allowedWhat would you do as:Player A: Player B: Player C: What would be the resulting outcome? (Who would survive?)4) Player A acts first, then B & C act simultaneously – a shot must be fired and no missing is allowedWhat would you do as:Player A: Player B: Player C: What would be the resulting outcome? (Who would survive?)5) Player A acts first, then B & C act simultaneously – a shot must be fired but missing is allowedWhat would you do as:Player A: Player B: Player C: What would be the resulting outcome? (Who would survive?)6) Player A acts first, then B & C act simultaneously – one can choose not to fire and missing is allowedWhat would you do as:Player A: Player B: Player C: What would be the resulting outcome? (Who would survive?) ................
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