# Parallel Repetition: Simplifications and the No-Signaling Case

## Thomas Holenstein

```
```In a two-player refereed game, a referee chooses (x,y) according to a
publicly known distribution P_XY, sends x to Alice, and y to Bob.
Without communicating with each other, Alice responds with a value ä"
and Bob responds with a value "b". Alice and Bob jointly win if a
publicly known predicate Q(x,y,a,b) holds.
Let such a game be given and assume that the maximum probability that
Alice and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998) shows
that if the game is repeated n times in parallel, then the probability
that Alice and Bob win all games simultaneously is at most
v'^(n/log(s)), where s is the maximal number of possible responses
from Alice and Bob in the initial game, and v' is a constant depending
only on v.
In this work, we simplify Raz's proof in various ways and thus shorten
it significantly. Further we study the case where Alice and Bob are
not restricted to local computations and can use any strategy which
does not imply communication among them.