### Table 6: A cyclic game with a unique Nash equilibrium

"... In PAGE 22: ... An M2 rule speci es a probability distribution over at most 2m possible actions by conservative and adaptive rules and assigns a best response to it. While the number of pure-action best responses to mixed actions may exceed 2m, one can construct a stage game similar to Table6 to restrict them to 2m as well. Any higher level mixed rule can be also restricted to have beliefs with nite support.... ..."

### Table 3: Unique Correlated Equilibria

"... In PAGE 15: ... The European tax-based approach (ET) is the more restrictive for the trading sectors in Germany and the United Kingdom but not in Italy and the other EU countries. 0 20 40 60 80 100 120 140 DEU UK ITA REU Emissions quo tas (M tC) GF DT HE ET Figure 4: Emission Allowances by Region Under Different Allocation Rules Table3 shows that the games have a unique correlated equilibrium, and that the equilibria are always different from the competitive equilibrium solutions, where the dif- ferent regions would play the same DT strategy. Germany, which is the main supplier of emission allowances, tends to rely on the domestic tax-based approach (DT) but the other regions, which are permits buyers, have an incentive to depart from this approach to... ..."

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### Table 3: Unique Correlated Equilibria

"... In PAGE 11: ...20 40 60 80 100 120 140 DEU UK ITA REU Emissions quo tas (M tC) GF DT HE ET Figure 4: Emission Allowances by Region Under Different Allocation Rules Table3 shows that the games have a unique correlated equilibrium, and that the equilibria are always different from the competitive equilibrium solutions, where the dif- ferent regions would play the same DT strategy. Germany, which is the main supplier of emission allowances, tends to rely on the domestic tax-based approach (DT) but the other regions, which are permits buyers, have an incentive to depart from this approach to maximize their own payoffs.... ..."

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### Table 3.3: Minimal Winning and Unique Tying Coalitions of Sample Game

2003

### Table 1: The components available in the game.

2004

"... In PAGE 8: ... There are four different components including CPU, motherboard, memory, and hard drive. Each com- ponent, except the CPU, comes in two different specifications, one high end and one low end, Table1 . The CPU comes in four unique products but with two different brands.... ..."

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### Table 8: Iterated Backward Inference in a matrix game tree for the Centipede game(seeFigure 3)

2003

"... In PAGE 20: ... This result agrees with standard backward induction; in Section 8 I establish that Iterated Backward Inference agrees with backward induction in all perfect information games with a unique subgame-perfect equilibrium, of which the Centipede game is an example. Table8 shows that Iterated Backward Inference eliminates just one strategy in the matrix tree (normal form) of the Centipede game, namely ll which is weakly dominated by lt.Afterll is dominated, there is no strict dominance, and the procedure terminates.... ..."

### Table 2: Player model for the 2 player game with coalitions.

"... In PAGE 2: ... The total genotype length is therefore 221 + 21 bits. 3 Evolving Coalition of Strategies Each player in the IPD game can be defined by the model given in Table2 . It describes all the information available to the player in deciding hidher next move.... In PAGE 2: ... It describes all the information available to the player in deciding hidher next move. In the Table2 , ID is assigned to each agent and unique identifier, and History is a memory to remember previous moves. Strategy is a lookup table for his next move.... ..."

### Table 2: Player model for the 2 player game with coalitions.

"... In PAGE 2: ... The total genotype length is therefore 22l + 2l bits. 3 Evolving Coalition of Strategies Each player in the IPD game can be defined by the model given in Table2 . It describes all the information available to the player in deciding his/her next move.... In PAGE 2: ... It describes all the information available to the player in deciding his/her next move. In the Table2 , ID is assigned to each agent and unique identifier, and History is a memory to remember previous moves. Strategy is a lookup table for his next move.... ..."

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### Table 1a: A coordination game. Table 1b: The (reduced) normal form of the 2-period game of endogenous timing associated with the game from Table 1a.

1993

"... In PAGE 6: ... Speci cally, the following example demonstrates that even if a game has a strict equilibrium that is also the unique Stackelberg equilibrium outcome in each of the two games where one of the players is forced to move rst, it is by no means obvious that players will necessarily coordinate on this equilibrium when the order of the moves is determined endogenously. Consider the common interest coordination game g given in Table1 a. When players have to move simultaneously, there are three equilibria: (T; L); (B; R) and a mixed equi- librium that yields each player the payo 2 3.... In PAGE 7: ...not commit he might be in three di erent information sets in period 2 and in each of these he can choose between two possible actions.) Table1 b displays a reduced form of this normal form: It only lists those strategies in which a player plays a best response in the second period whenever the opponent has unilaterally committed himself in the rst period. Since this reduction makes it easier to see the intuition for our results, our discussion in this section will be based on it.... In PAGE 7: ... Note, however, that our formal results are based on the full, unreduced, game. In Table1 b, X1 denotes the pure strategy \commit to X in period 1 quot;, while X2 denotes \wait till period 2, play the unique best response if the opponent has already moved in period 1 and, otherwise, play X. quot; The question we want to address is whether \2,2 quot; (i.... In PAGE 7: ...e. the commitment robust equilibrium) is the unique \sensible quot; outcome in the game of Table1 b? Alternatively, can one perhaps also give good arguments in favor of \1,1 quot;, or in favor of the mixed equilibrium? Inspection shows that the game from Table 1b has three Nash equilibrium outcomes: There is a connected set of equilibria with payo (2,2), there is also a connected set of equilibria with payo (1,1) and there is a completely mixed equilibrium in which each player plays (1,2,4,8)/15. The latter yields each player the payo 14/15.... In PAGE 7: ...e. the commitment robust equilibrium) is the unique \sensible quot; outcome in the game of Table 1b? Alternatively, can one perhaps also give good arguments in favor of \1,1 quot;, or in favor of the mixed equilibrium? Inspection shows that the game from Table1 b has three Nash equilibrium outcomes: There is a connected set of equilibria with payo (2,2), there is also a connected set of equilibria with payo (1,1) and there is a completely mixed equilibrium in which each player plays (1,2,4,8)/15. The latter yields each player the payo 14/15.... In PAGE 7: ...oordinate on (T; L) or (B; R), i.e. I prevent a disequilibrium outcome. However, if I wait, my opponent does better by committing to his Stackelberg leader strategy. Having eliminated the mixed equilibrium from Table1 a, let us now turn to the Pareto dominated pure strategy equilibrium \1,1 quot;. A glance at Table 1b shows that this is not as easily eliminated.... In PAGE 7: ... Having eliminated the mixed equilibrium from Table 1a, let us now turn to the Pareto dominated pure strategy equilibrium \1,1 quot;. A glance at Table1 b shows that this is not as easily eliminated. Namely, B1 and R1 are undominated strategies in Table 1b so that (B1; R1) is a perfect equilibrium.... In PAGE 7: ... A glance at Table 1b shows that this is not as easily eliminated. Namely, B1 and R1 are undominated strategies in Table1 b so that (B1; R1) is a perfect equilibrium. Even stronger, (B1; R1) is a proper equilibrium (Myerson (1978)): For any quot; gt; 0 that is su ciently small the strategy pair in which each player plays the completely mixed strategy ( quot;3; quot;2; quot;; 1 ? quot; ? quot;2 ? quot;3) is a 2 quot;-proper... In PAGE 7: ... Namely, B1 and R1 are undominated strategies in Table 1b so that (B1; R1) is a perfect equilibrium. Even stronger, (B1; R1) is a proper equilibrium (Myerson (1978)): For any quot; gt; 0 that is su ciently small the strategy pair in which each player plays the completely mixed strategy ( quot;3; quot;2; quot;; 1 ? quot; ? quot;2 ? quot;3) is a 2 quot;-proper equilibrium in Table1... In PAGE 8: ...) Intuitively, if players expect mistakes to occur with a relatively large probability in the second period, or if they believe that the unattractive mixed strategy equilibrium would be played in case the second period would be reached, then each player has a strong incentive to commit to the equilibrium \1,1 quot; in the rst period if he expects his opponent to do the same. Hence, it seems that Rosenthal apos;s (1991) conclusion that only (T; L) makes sense in the game of Table1 a when aspects of commitment are taken into account was premature. The concept of Nash equilibrium allows two interpretations.... In PAGE 8: ...T; L). Let us illustrate this result for curb*-equilibria, i.e. for equilibria that belong to minimal sets of mixed strategy pairs that are closed with respect to taking undominated best responses.1 The game of Table1 b has two sets that are closed under (undominated) best responses, viz. the entire strategy set and the set of strategies in which zero proba- bility is assigned to B1 and R1.... In PAGE 8: ... Obviously, only the latter set is minimal. Furthermore, each equilibrium in this set induces the outcome \2,2 quot;, hence, each curb* equilibrium of the game of Table1 b produces the commitment robust equilibrium of Table 1a. In Section 6 we show that the above argument generalizes to any game that admits a unique 1Since we here consider the reduced game, in this section this result holds also for curb-equilibria, i.... In PAGE 10: ... It is easy to construct games without any CRE, take the Battle of the Sexes, for ex- ample. On the other hand, the set of games with a pure CRE is certainly not of measure zero: All games that are close to the game from Table1 a have (T; L) as a pure CRE. Two classes of games that do admit CRE are the strictly competitive games (Friedman... In PAGE 26: ... It is worthwhile to point out that this result depends essentially on the restriction to pure strategies. Namely, consider the game from Table1 a and assume that moving in period 1 costs quot; gt; 0. Now (B1; R1) is no longer an equilibrium, but there is a mixed equilibrium close to it, namely each player choosing (0; quot;; 2 quot;; 1?3 quot;).... ..."

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