Alphabet Puzzles (Grades K-1)

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A set of players, I = {1, . . , I }. 2. A set H of sequences, referred to as histories, defined as follows: h0 = ∅ s 0 = (s10 , . . , sI0 ) h1 = s 0 .. initial history stage 0 action profile history after stage 0 .. hk+1 = (s 0 , s 1 , . . , s k ) history after stage k k = be the set of all possible stage k histories. Then, H = ∪∞ Let k=0 H is the set of all possible histories. If the game has a finite number (K + 1) of stages, then it is a finite horizon game. We use H K+1 to denote the set of all possible terminal histories.

For all s in the support of σ ∗ (combined with the preceding We first show that Ei∗ = ui (si , σ−i i i relation, this proves one implication). Assume to arrive at a contradiction that this is not the case, ∗ ) < E ∗ . , there exists an action si in the support of σi∗ such that ui (si , σ−i i i −i i i for all si ∈ Si , this implies that si ∈Si ∗ σi∗ (si )ui (si , σ−i ) < Ei∗ , – a contradiction. The proof of the other implication is similar and is therefore omitted. ✷ It follows from this characterization that every action in the support of any player’s equilibrium mixed strategy yields the same payoff.

31 Glicksberg Every continuous game has a mixed strategy Nash equilibrium. With continuous strategy spaces, the space of mixed strategies is infinite-dimensional; therefore, we need a more powerful fixed point theorem than the version of Kakutani we have used before. Here we adopt an alternative approach to prove Glicksberg’s Theorem, which can be summarized as follows: • We approximate the original game with a sequence of finite games, which correspond to successively finer discretizations of the original game.

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