The definition uses preference relations instead of utility functions. This is to accommo- date payoff aggregation criteria which do not admit an expected-utility representation I will only illustrate the key features of the two "special"criteria proposed in OR, and then focus on discounting. I will mostly discuss infinitely-repeated games Discounting In general, we assume that players share a common discount factor d E(0, 1), and rank payoff streams according to the rule (n21x;(m2)1∑6-1(n2-2) Now, we want to be able to talk about payoff profiles of the stage game G as being achieved in the repeated version of G. Thus, for instance, if the profile a E A is played in each repetition of the game, we want to say that the payoff profile u(a) is attained. But, of course, this is not the discounted value of the stream (u(a), u(a),. ) rather, this value is Thus, one usually“ Measures"” payoffs in a repeated game in“ discounted units”,i.e.in terms of the discounted value of a constant payoff stream(1,1,.), which is of course s The bottom line is that, given the terminal history(a, .. ar, . .) the correspondin payoff profile is taken to be(1-d>t18-u(a) Limit -of-Means The key feature(OR: " the problem")of discounting is that periods are weighted differently One might think that, in certain situations, this might be unrealistic: OR propose the example of nationalist struggles A criterion which treats period symmetrically is the limit of means. Ideally, we would want the value of a stream(u)>1 to Player i to be limT- EI,. However, this limit may fail to exist. I Thus, we consider the following rule 21分 lim inf-t T 0 Recall how lim infn In is defined: let yn= inf(m: m 2 n), and set lim inf, n= limn yn Thus, lim infn In>0 iff, for some e>0, In>E for all but finitely many n's Here' s how to construct an example: call v t the average up to T. First, set u l= 1, so Vt=1 Now consider the sequence(1,0,0): V3=1. Now extend it to(1,0, 0, 1, 1, 1): V6 (1,0,0, 1, 1, 1,0,0,0,0,0,0):V2=3. You see how you can construct a sequence such that the sequence of verges fips between 3 and 3The definition uses preference relations instead of utility functions. This is to accommo￾date payoff aggregation criteria which do not admit an expected-utility representation. I will only illustrate the key features of the two “special” criteria proposed in OR, and then focus on discounting. I will mostly discuss infinitely-repeated games. Discounting In general, we assume that players share a common discount factor δ ∈ (0, 1), and rank payoff streams according to the rule (u t i )t≥1 i (w t i )t≥1 ⇔ X t≥1 δ t−1 (u t i − w t i ) > 0 Now, we want to be able to talk about payoff profiles of the stage game G as being achieved in the repeated version of G. Thus, for instance, if the profile a ∈ A is played in each repetition of the game, we want to say that the payoff profile u(a) is attained. But, of course, this is not the discounted value of the stream (u(a), u(a), . . .): rather, this value is u(a) 1−δ . Thus, one usually “measures” payoffs in a repeated game in “discounted units”, i.e. in terms of the discounted value of a constant payoff stream (1, 1, . . .), which is of course 1 1−δ . The bottom line is that, given the terminal history (a 1 , . . . , at , . . .), the corresponding payoff profile is taken to be (1 − δ) P t≥1 δ t−1u(a t ). Limit-of-Means The key feature (OR: “the problem”) of discounting is that periods are weighted differently. One might think that, in certain situations, this might be unrealistic: OR propose the example of nationalist struggles. A criterion which treats period symmetrically is the limit of means. Ideally, we would want the value of a stream (u t i )t≥1 to Player i to be limT→∞ PT t=1 u t i T . However, this limit may fail to exist.1 Thus, we consider the following rule: (u t i )t≥1 i (w t i )t≥1 ⇔ lim inf t→∞ X T t=1 u t i − w t i T > 0 [Recall how lim infn xn is defined: let yn = inf{xm : m ≥ n}, and set lim infn xn = limn yn. Thus, lim infn xn > 0 iff, for some  > 0, xn >  for all but finitely many n’s.] 1Here’s how to construct an example: call V t i the average up to T. First, set v 1 i = 1, so V t i = 1. Now consider the sequence (1,0,0): V 3 i = 1 3 . Now extend it to (1,0,0,1,1,1): V 6 i = 2 3 . Now consider (1,0,0,1,1,1,0,0,0,0,0,0): V 1 i 2 = 1 3 . You see how you can construct a sequence such that the sequence of averages flips between 1 3 and 2 3 . 2