Which was the first Sci-Fi story to predict obnoxious "robo calls"? This process continues until no more strategies can be deleted. /Filter /FlateDecode f@n8w3jbx|>,cMm[6Rii6n^c3.9ed(Wq[)9?YrM\;Xdoo}#Jlyjs9a9?oq>VRbErX0 We can push the logic further: if Player 1 knows that Player 2 is . Awesome!! By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. strategies surviving iterative removal of strictly dominated strategies. $u_1(U,x) = 5-4(a+b)$, $u_1(M,x) = 1$, $u_1(B,x) = 1+4a$. Iterated elimination of strictly dominated strategies cannot solve all games. is there such a thing as "right to be heard"? 12 0 obj /Matrix [1 0 0 1 0 0] 1,1 & 1,5 & 5,2 \\ Built In is the online community for startups and tech companies. Recall from last time that a strategy is strictly dominated if another strategy exists that always pays strictly more regardless of what other players are doing. 33 0 obj << Share. We can set a mixed strategy where player 1 plays up and down with probabilities (,). Are all strategies that survive IESDS part of Nash equilibria? How can I control PNP and NPN transistors together from one pin? This limits the usefulness of this solution concept. If column mixes over $(L, M)$ - $x = (a, 1-a, 0)$ We obtain a new game G 1. In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. The iterated deletion of dominated strategies is one common, but tedious, technique for solving games that do not have a strictly dominant strategy. This page was last edited on 30 March 2023, at 12:02. /FormType 1 Weak Dominance Deletion Step-by-Step Example: In any case, if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a dominance-solvable game. For any possible strategy by Bar As opponent, there is some strategy that gives higher payoff than the $2 strategy. QUEby``d34zJ$82&q?n30 BK$fG-9F!84IsP\E^|Tr"4~0'.t[q5iPM2,^)0-]1(hVY~ O9dgO8u pD%] l['qVa4R3v+nrgf9#'Lt^044Q@FkoB3R=hHe+}];s\!@9MHLi{ >> endobj Some strategiesthat were not dominated beforemay be dominated in the smaller game. 28 0 obj tar command with and without --absolute-names option. . /ProcSet [ /PDF ] endstream Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Dominated Strategies & Iterative Elimination of Dominated Strategies 3. For the row player R the domination between strategies can be seen by comparing the rows of the matrices P R. xWKo6W:K6h^g,)PofHJ0iH`d=`De0 For Bar A, there is no price that will give it higher revenues than any other price it could have set, no matter what price Bar B sets. In this case, we should eliminate the middle strategy for player 1 since its been dominated by the mixed strategy of playing up and down with probability (,). % Sorted by: 2. density matrix, English version of Russian proverb "The hedgehogs got pricked, cried, but continued to eat the cactus". /Shading << /Sh << /ShadingType 3 /ColorSpace /DeviceRGB /Domain [0 1] /Coords [4.00005 4.00005 0.0 4.00005 4.00005 4.00005] /Function << /FunctionType 2 /Domain [0 1] /C0 [0.5 0.5 0.5] /C1 [1 1 1] /N 1 >> /Extend [true false] >> >> The row player's strategy space is $(U,M,B)$ and the column palyer's is $(L,M,R)$. Enter type of game: General m x n game (A,B) Zerosum m x n game (A,-A) Symmetric m x m game (A,AT) For zerosum and symmetric games, only enter payoff matrix A for player 1. << /S /GoTo /D (Outline0.4) >> Can my creature spell be countered if I cast a split second spell after it? xP( For any possible strategy by Bar As opponent, there is some strategy that gives higher payoff than the $2 strategy. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. We can apply elimination of -dominated strategies iteratively, but the for Can I use my Coinbase address to receive bitcoin? In this sense, rationalizability is (weakly) more restrictive than iterated deletion of strictly dominated strategies. The first (and preferred) version involves only eliminating strictly dominated strategies. Stall Wars: When Do States Fight to Hold onto the StatusQuo? The game is symmetric so the same reasoning holds for Bar B. The best answers are voted up and rise to the top, Not the answer you're looking for? By the well known path independence of iterated elimination of strictly dominated strategies [1, 19, 41], fully reducing and results in the same game. The spreadsheet works very well and congratulations.I really do not know why the guy Cogito is claimming about. 34 0 obj << Lets look at the strategy profile ($2, $5). x}V[7SHQu'X6Yjuf`a5IG*YR|QRJz?uhn~~}?Ds&>y: I only found this as a statement in a series of slides, but without proof. Here is a quick Python implementation for . endobj The answer is positive. >> Want to practice what Im learning, and as far as I can find your calculator seems to be the only easiest best option available. We can generalize this to say that rational players never play strictly dominated strategies. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.). Nash Equilibrium Dominant Strategies Astrategyisadominant strategy for a player if it yields the best payo (for that player) no matter what strategies the other players choose. appreciated tremendously! stream I only found this as a statement in a series of slides, but without proof. This is called Strictly Dominant Mixed Strategies. Q: (2) Consider the following two-player norma. Consider the following game to better understand the concept of iterated elimination of strictly dominated strategies. 1 Answer. I am jumping back into this after almost 20 years,,, with John Maynard Smiths Evolution and the Theory of Games. Player 1 has two strategies and player 2 has three. /Type /XObject Many simple games can be solved using dominance. (Iterated Delation of Strictly Dominated Strategies) Change). stream That is, if a strategy is strictly dominated, it can't be part of a Nash equilibrium. cZiAIF}$\ScQME What are the pure strategy Nash equilibria (PSNE)? To solve the games, the method of iterated elimination of strictly dominated strategies has been used. /Length 15 Pricing at $5 would be. A: Pure strategy nash equilibrium is the one in which all the players are doing their best, given the. [2], Common Knowledge: The assumption that each player has knowledge of the game, knows the rules and payoffs associated with each course of action, and realizes that every other player has this same level of understanding. The argument for mixed strategy dominance can be made if there is at least one mixed strategy that allows for dominance. >> endobj A dominant strategy in game theory occurs when one player has a stronger, more effective strategy over another player. Games between two players are often . No. It involves iteratively removing dominated strategies. &BH 6a}F~DB ]%pg BZ8PT LAdku|u! What is this brick with a round back and a stud on the side used for? Doubling Down: The Dangers of Disclosing SecretActions, Getting a Hand By Cutting Them Off: How Uncertainty over Political Corruption AffectsViolence, How Fast and How Expensive? Conversely, a strategy is dominated if it leads a player to worse outcomes than . and an additional point for being at their preferred entertainment. As a result, the Nash equilibrium found by . The calculator works properly, at least in the case you brought to my attention. /Length 4297 Then you can reason that I will not play something because you know that I can reason that you will not play something. So the NE you end up with is $(T,L)$. When player 2 plays left, then the payoff for player 1 playing the mixed strategy of up and down is 1, when player 2 plays right, the payoff for player 1 playing the mixed strategy is 0.5. For player 2, however, right is /Type /Page /Parent 17 0 R I know that Iterated Elimination of Strictly Dominated Strategies (IESDS) never eliminates a strategy which is part of a Nash equilibrium. B & 2, -2 & 1, -1 & -1, -1 Nash-equilibrium for two-person zero-sum game. %PDF-1.4 It is possible that an action is not strictly dominated by any pure strategy, but strictly dominated by a mixed strategy. For instance, consider the payoff matrix pictured at the right. S1= {up,down} and S2= {left,middle,right}. 6D7wvN816sIM" qsG;!_maeq"Mw]Vn1cJf}?!!u"\W,v,hTc}yZoV]}_|u_F+tA@1g(,* ^ZR~@Om8eY Oqy*&C3FW1J"&2Nm*z}y}^ a6`wC(=h:*4"0xSdgE+;>ef,XV> W*8}'n~oP> Strategy: an introduction to game theory (Second ed.). 6.3. This is an Excel spreadsheet that solves for pure strategy and mixed strategy Nash equilibrium for 22 matrix games. 5m_w:.A:&Wvg+1c (see IESDS Figure 1). I finished my assignment with the help of those, and just checked my answers on your calculator I got it right! For example, a game has an equilibrium in dominant strategies only if all players have a dominant strategy. Mean as, buddy! stream Thus if player 1 knows that player 2 is rational then player 1 can & L & C & R \\ \hline The second applet considers 2x2 bi-matrices. M. We now focus on iterated elimination of pure strategies that are strictly dominated by a mixed strategy. %PDF-1.5 1,2 & 1,1 & 1,1 \\ 1 0 obj << This results in a new, smaller game. I could find the equations on wikipedia, for the love of god. Strictly dominated strategies cannot be played in equilibrium, and you will note that the calculator says that is the PSNE. /Resources << \begin{array}{c|c|c|c} What were the poems other than those by Donne in the Melford Hall manuscript? IESDS on game with no strictly dominated strategies. 31 0 obj << 50 0 obj << Locals will buy from the bar setting the lowest price (and will choose randomly if the two bars set the same price). Why did DOS-based Windows require HIMEM.SYS to boot? Thinking about this for a moment, a follow up . Problem 4 (30 points). There are two versions of this process. T & 2, 1 & 1, 1 & 0, 0 \\ \hline better than up if 2 plays right (since 2>0). In 2-player games, the strategies that survive iterated elimination of strictly dominated strategies are called rationalizable. /PTEX.InfoDict 51 0 R The first step is repeated, creating a new, even smaller game, and so on. And is there a proof somewhere? How can I control PNP and NPN transistors together from one pin? Mixed-strategy Nash equilibrium. /FormType 1 The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play. Ive used a lot of terminology, so lets look at an example to clarify these concepts. So, thank you so much! After all, there are many videos on YouTube from me that explain the process in painful detail. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. That is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum. /Matrix [1 0 0 1 0 0] /Filter /FlateDecode However, that Nash equilibrium is not necessarily "efficient", meaning that there may be non-equilibrium outcomes of the game that would be better for both players. : Whereas looking for an equilibrium in strictly dominant strategies involves finding a strategy that is always the best response for each player, looking for an equilibrium via iterated deletion involves iteratively discounting from consideration strategies that are never best responses. Sorry!) IESDS on game with no strictly dominated strategies. The actions surviving the iterated elimination of strictly dominated strategies are not de-pendent on the exact sequence of elimination. We will have to broaden our solution concept if we want to make progress elsewhere. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique, Two bars, Bar A and Bar B, are located near each other in the city center. by making M the new strictly dominant strategy for each player. (Dominant and Dominated Strategies) /#)8J60NVm8uu_j-\L. Id appreciate it if you gave the book a quick review over on Amazon. (=. >> It turns out that in 2-player games, the two concepts . Once weve identified the players and the strategies, we can begin to create our payoff matrix: Now, we can fill in the payoffs. rev2023.4.21.43403. So, we can delete it from the matrix. endstream A best . /Font << /F45 4 0 R /F50 5 0 R /F46 6 0 R /F73 7 0 R /F15 8 0 R /F27 9 0 R /F28 10 0 R /F74 11 0 R /F76 12 0 R /F25 13 0 R /F32 14 0 R /F62 15 0 R /F26 16 0 R >> Consequently, if player 2 knows that player 1 is rational, and player 2 Does a password policy with a restriction of repeated characters increase security? Tourists will choose a bar randomly in any case. Unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. Okay, thanks, now I understand. Try watching this video on. The first step is repeated, creating a new even smaller game, and so on. we run into many situations where certain issues are bookend policies (0 or 1), but for which one side has a distribution of options that can be used to optimize, based on previous decisions made using such policies (a priori info from case studies). So playing strictly dominant strategies is Pareto e cient in the \no-talking norm"-modi ed PD. strategies. There are instances when there is no pure strategy that dominates another pure strategy, but a mixture of two or more pure strategies can dominate another strategy. That is, when Bar A charges $2 and Bar B charges $5. eH\h GPqq rDn%,p;/K0 Jb{Cx3vmQ6JX4|qXhxL` bF$9 "5v'2WuGdBmq+]-m>ExV#3[2Z9'hxOpT, ^.\K|Z.+G%IOIB h "FtMUvr! z$"xh~w{e` /Filter /FlateDecode >> Why do men's bikes have high bars where you can hit your testicles while women's bikes have the bar much lower? As weve seen, the equilibrium dominated strategies solution concept can be a useful tool. $)EH (Note: If there are infinitely many equilibria in mixed strategies, it will not calculate them. endobj Step 1: B is weakly dominated by T. Step 2: R is weakly dominated by C. Step 3: C is weakly dominated by L. Step 4: M is weakly dominated by T. So the NE you end up with is ( T, L). The applet calculates . For player 1, neither up nor down is strictly dominated. >> For this method to hold however, one also needs to consider strict domination by mixed strategies. This also satisfies the requirements of a Nash equilibrium. << /S /GoTo /D (Outline0.2) >> Recall IDSDS is Iterated Deletion of Strictly Dominated Strategies and ID-WDS is Iterated Deletion of Weakly Dominated Strategies Proposition 1 Any game as at most one weakly dominant solution. But I can not find any weakly dominated strategy for any player. The predictive power may not be precise enough to be useful. Each bar has 60 potential customers, of which 20 are locals. $$ A B () Pay Off . bm'n^ynC-=i)yJ6#x,rcTHHNYwULy2:Mjw'jjn!C}<4C[L,HO[^#B>9Fam%'QvL+YN`LRoOrD{G%}k9TiigB8/}w q#Enmdl=8d2 (o BmErx `@^PB2#C5h0:ZM[L,x4>XLHNKd88(qI#_kc&A's ),7 'beO@nc|'>E4lpC Very cool! tation in few rounds of iterated elimination of strictly-dominated strategies. Thanks! Proof The strategy a dominates every other strategy in A. ;UD(`B;h n U _pZJ t \'oI tP*->yLRc1,[j11Y(25"1U= 3 0 obj << If you cannot eliminate any strategy, then all strategies are rationalizable. not play right. Recall from last time that a strategy is strictly dominated if another strategy exists that always pays strictly more regardless of what other players are doing. Iterated Elimination of Weakly Dominated Strategies with Unknown Parameters. 5,1 & 1,5 & 1,2 \\ This game can easily be solved by iterated elimination of strictly dominated strategies, yielding the prole (D;R;A). To find the unique surviving solution, we use the Iterated Elimination of . In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies. Elimination of weakly dominated strategies - example, Improving the copy in the close modal and post notices - 2023 edition, New blog post from our CEO Prashanth: Community is the future of AI, Reduce the payoff matrix using (weakly) dominated strategies. The first thing to note is that neither player has a dominant strategy. Ther is no pure Nash equilibrium if where the row player plays $M$, because column's best response is $U$, but to $U$ row's best response ins $B$. Did the Golden Gate Bridge 'flatten' under the weight of 300,000 people in 1987. E.g., cash reward, minimization of exertion or discomfort, promoting justice, or amassing overall utility - the assumption of rationality states that Do Nonproliferation AgreementsConstrain? 5,1 & 1,5 & 1,2 \\ Examples. /Type /XObject We can then fill in the rest of the table, calculating revenues in the same way. This is the single Nash Equilibrium for this game. /Length 15 In this scenario, for player 1, there is no pure strategy that dominates another pure strategy. Some strategies that werent dominated before, may be dominated in the smaller game. For player 1, neither up nor down is strictly dominated. The reason it lists strictly dominated strategies instead of strictly dominant strategies is that there is no guarantee that a player will play a strictly dominant strategy in equilibrium once you extend past 22 matrices. (up,middle) as the outcome of the game. It is well known |see, e.g., the proofs in Gilboa, Kalai, and Zemel (1990) and Osborne and Rubinstein (1994)| that the order of elimination is irrelevant: no matter which order is used, $u_1(U,x) = 5-4a$, $u_1(M,x) = 1$, $u_1(B,x) = 1$. More generally, the strategies that remain after a process of iterated deletion of strictly dominated strategies are known as rationalizable strategies. We cannot delete anything else. Im not the first person to say this as evidenced above but without your YouTube lessons I would be struggling through my second-year game theory course. . A player has a dominant strategy if that strategy gives them a higher payoff than anything else they could do, no matter what the other players are doing. Could a subterranean river or aquifer generate enough continuous momentum to power a waterwheel for the purpose of producing electricity? a weakly dominant strategy is a strategy that provides at least the same utility for all the other player's strategies, and strictly greater for some strategy. 15 0 obj >> The logic of equilibrium in dominant strategies is that if a player has a strategy that is always best, we would expect him to play it. 49 0 obj << Now let us put ourselves in the shoes of Bar A again. Up is better than down if 2 plays left (since 1>0), but down is iuO58QG*ff/Uajfk@bogxeXNA 3eE`kT,~u`y)2*Amsgqm#0Py7N7ithA7@z|O:G#`IFR1Zwzdz: y[ i+8u#rk3)F@E[3r(xz)R2O{rhM! \end{array} /BBox [0 0 5669.291 8] endobj Strategy: A complete contingent plan for a player in the game. In this scenario, the blue coloring represents the dominating numbers in the particular strategy. A dominated strategy in game theory occurs when one player has a more dominant strategy over another player. Wouldn't player $2$ be better off by switching to $C$ or $L$? /Filter /FlateDecode I have attached a 2003 version to the original post, but not guarantees it functions properly. Player 1 has two strategies and player 2 has three. (Dominated strategy) For a player a strategy s is dominated by strategy s 0if the payo for playing strategy s is strictly greater than the payo for playing s, no matter what the strategies of the opponents are. /Contents 3 0 R pruning of candidate strategies at the cost of solu-tion accuracy. Assuming you cannot reduce the game through iterated elimination of strictly dominated strategies, you are basically looking at taking all possible combinations of mixed strategies for each player and seeing if an opposing strategy can fulfill the Nash conditions. %w`T9:?H' ^mNA\4" . Sorry I wrote the answer on my phone. You said in your video that down-right was the strictly dominated strategy, but your excel spreadsheet says top left is. Thus regardless of whether player 2 chooses left or right, player 1 gets more from playing this mixed strategy between up and down than if the player were to play the middle strategy. It also ensures that there is a strictly dominant strategy pro le s 2S satisfying u i(s ) > u i(s) for all i 2N and all s 2S satisfying s 6= s . D +(91)-9821210096 | paula deen meatloaf with brown gravy. A: As we answer only 3 subparts . 2For instance, in some extensive games, backward induction may be an elimination order of condition-ally dominated strategies that is not maximal, as will be shown in Example 2. I am particularly interested in the ideas of honesty, bargaining, and commitment as these factor strongly in decision making in multi-stakeholder groups e.g., where bargaining/haggling/negotiating produces commitments. We can demonstrate the same methods on a more complex game and solve for the rational strategies. endobj strictly. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. This gives Bar B a total of 20 beers sold at a price of $5 each, or $100 in revenue. I know that Iterated Elimination of Strictly Dominated Strategies (IESDS) never eliminates a strategy which is part of a Nash equilibrium. Therefore, Bar A would never play the strategy $2. $u_1(U,x) > u_1(M,x) \wedge u_1(B,x) > u_1(M,x) \Rightarrow$ if column plays x row plays $M$ with probability zero. Unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. (LogOut/ There are two versions of this process. As in Chapter 3 we would like to clarify whether it aects the Nash equilibria, in this case equilibria in mixed strate-gies. Fortunately, there is a solution concept that does guarantee to return a tractably small set of expected outcomes known as the Nash equilibrium. If Player 2 chooses U, then the final equilibrium is (N,U). /FormType 1 ^qT4ANidhu z d3bH39y/0$ D-JK^^:WJuy+,QzU.9@y=]A\4002lt{ b0p`lK0zwuU\,(X& {I 5 xD]GdWvM"tc3ah0Z,e4g[g]\|$B&&>08HJ.8vdN.~YJnu>/}Zs6#\BOs29stNg)Cn_0ZI'9?fbZ_m4tP)v%O`1l,>1(vM&G>F 5RbqOrIrcI5&-41*Olj\#u6MZo|l^,"qHvS-v*[Ax!R*U0 If Bar B is expected to play $5, Bar A can get $80 by playing $2 also and can get $160 by playing $4. It uniquely survives the iterated elimination of strictly dominated strategies, so the unique Nash equilibrium for this case is (Row k+1, Column k+1). (: dominant strategy) "" ("") (: dominance relation) . In that case, pricing at $4 is no longer Bar As best response. In the prisoners dilemma, up and left (cooperate for the players) are strictly dominated. Therefore, Player 2 will never play strategy Z. depicted below. Rational players will never use such strategies. Iterated elimination of strictly dominated strategies (IESDS). In fact, the logic can grow more complicated. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. It seems like this should be true, but I can't prove it myself properly. 24 0 obj The expected payoff for playing strategy X + Z must be greater than the expected payoff for playing pure strategy X, assigning and as tester values. Some authors allow for elimination of strategies dominated by a mixed strategy in this way. /Subtype /Form The result of the comparison is one of: This notion can be generalized beyond the comparison of two strategies. >> However, neither of these methods is guaranteed to return a tractably small set of expected outcomes. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium, referred to as a "dominant strategy equilibrium". A straightforward example of maximizing payoff is that of monetary gain, but for the purpose of a game theory analysis, this payoff can take any desired outcome. Q: Address the following with suitable examples. Player 1 knows he can just play his dominant strategy and be better off than playing anything else. ngWGNo Choose a player and remove all the strictly dominated strategies for that player. $u_1(U,x) > u_1(M,x) \wedge u_1(B,x) > u_1(M,x) \Rightarrow$ if column plays x row plays $M$ and $B$ with probability zero. << /S /GoTo /D (Outline0.1) >> /ProcSet [ /PDF ] /Resources 1 0 R Does the 500-table limit still apply to the latest version of Cassandra? Accordingly, a strategy is dominant if it leads a player to better outcomes than alternative strategies (i.e., it dominates the alternative strategies). Watch on. Also, there are no strictly dominated strategies because a strictly dominated strategy cannot be a best response for any possible belief. With the dashed lines and the numbers beside them, we indicate the order of iterated elimination of conditional strictly dominated strategies. B:R>)`Q. I.e. Once I realized that I decided to ignore the application entirely. Untitled - Free download as PDF File (.pdf), Text File (.txt) or read online for free. And for column nothing can be eliminate anyway.). If Player 2 chooses T, then the final equilibrium is (N,T), O is strictly dominated by N for Player 1. Since in one case, one does better by playing C instead of D and never does worse, C weakly dominates D. Despite this, To apply the Iterated Elimination of Strictly Dominated Strategies (IESDS), we examine each row and column of the matrix to find strictly dominated strategies, i.e., those that always result in a lower payoff than another strategy regardless of the opponent's move. /k\MI\R}n%-(vvao5 %K6~hfmake/@v.6v]ko]cq"AI X4/F B{T% Of the remaining strategies (see IESDS Figure 4), Y is strictly dominated by X for Player 2. This follows from the earlier comment that a strictly dominated strategy is never a best response. So, is there any way to approach this? 1,1 & 1,5 & 5,2 \\ tar command with and without --absolute-names option. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. endobj Yes. 4 + 5 > 5 consideration when selecting an action.[2]. Two dollars is a strictly dominated strategy for Bar B, and Bar A knows this, too. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Each bar has 60 potential customers, of which 20 are locals and 40 are tourists. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. /Parent 47 0 R Embedded hyperlinks in a thesis or research paper. >> Only one rationalizable strategy is left {A,X} which results in a payoff of (10,4).

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