summaryrefslogtreecommitdiff
path: root/games
diff options
context:
space:
mode:
authorJeff Burdges <burdges@gnunet.org>2018-04-25 08:15:56 +0200
committerJeff Burdges <burdges@gnunet.org>2018-04-25 08:15:56 +0200
commitbad189ee87a5724a79622b0fe0e07ed3620fe068 (patch)
treebb3f73ef0e7ac0d0b7f650b1808df33fb2d964c8 /games
parentcf4086163362d68011d093bc683732bacac5aba1 (diff)
downloadpapers-bad189ee87a5724a79622b0fe0e07ed3620fe068.tar.gz
papers-bad189ee87a5724a79622b0fe0e07ed3620fe068.tar.bz2
papers-bad189ee87a5724a79622b0fe0e07ed3620fe068.zip
Fix income transperency
Diffstat (limited to 'games')
-rw-r--r--games/games.tex47
1 files changed, 28 insertions, 19 deletions
diff --git a/games/games.tex b/games/games.tex
index ebe551a..a609624 100644
--- a/games/games.tex
+++ b/games/games.tex
@@ -904,36 +904,45 @@ choice of $\gamma$, as otherwise the refresh always fails.
As our $\gamma$ are chosen randomly, any given refresh with one
false planchet has only a $1\over\kappa$ chance of succeeding.
+% We recall ... MAYBE COPY FROM REMOVED TEXT IN cf4086163362d68011d093bc683732bacac5aba1
If such a malicious refresh is detected, the coin becomes unspendable,
and contributes to the lost coins $b := w - s$
instead of the winnings $p := L - w'$ aka $|X|$.
We observe $b + p$ gives the value of refreshes attempted with
false planchets. Of course $p = 0$ and $b = 0$ if the adversary never
-uses false planchets, so no $R_C$ occur, leaving $p \over b + p$ undefined.
+uses false planchets, i.e. no $R_C$ occur, leaving $p \over b + p$ undefined.
-We shall prove $E({p \over b + p}) = {1\over\kappa}$ (\dag)
-with the expectation taken over games with false planchets.
-We obtain a valid game by terminating a game early, so
- we work by induction on the length of the game.
-
-% https://math.stackexchange.com/questions/852890/expectation-of-random-variables-ratio
-If we group together the games identical up until some $R_C$ with value $v$,
-then we find $R_C$
-...
-contributes $v\over\kappa$ to $p$ and $\kappa v$ to $b + p$.
-
-
- by grouping games identical up to an $R_C$
- according to the simulator's random choice of $\gamma$.
-Of course
We defined $R_C$ to be an anti-chain in the refresh forest, but
emphasize that an adversary who wins an $R_C$ should not gamble again,
-while an adversary who looses an $R_C$ looses $C$.
-It follows that each $R_C$ contributes ${1\over\kappa}$ towards the total expectation.
-{RETHINK WHAT HAPPENS HERE AND ABOVE}
+while an adversary who looses an $R_C$ looses $C$ completely.
+There is no way to influence $p$ or $b$ through withdrawals or spends
+by corrupted users of course. In principle, one could decrease $b$ by
+ sharing from a corrupted user to a non-corrupted users,
+but we may assume this does not occur either.
+
+We obtain a valid game run by terminating a game early, or even by
+blocking oracle calls on decedents of a particular coin $C$,
+ which we refer to as a strictly simpler game run.
+We shall prove $E({p \over b + p}) = {1\over\kappa}$ (\dag)
+with the expectation taken over games with false planchets
+ in which the adversary plays optimally
+ in that no strictly simpler game run increases $p \over b + p$.
+
+For any $R_C$,
+there are $\kappa$ game runs identical up through the commitment phase
+of $R_C$ that diverge in simulator's random choice of $\gamma$,
+ and blocks $C$ afterwards.
+If $v$ is the value of $R_C$ then one run adds $v$ to $p$, while
+ the remaining $\kappa-1$ runs add $v$ to $b$.
+
+An adversary may respond adaptively to $\gamma$ of course, but
+induction on the length of the game now still produces disjoint groupings
+$\mathbb{G}$ of optimal games in which
+ $\sum_{\mathbb{G}} {p \over b + p} = {1\over\kappa} |\mathbb{G}|$.
We conclude that $E({p \over b + p}) = {1\over\kappa}$, as desired.
\end{proof}
+% https://math.stackexchange.com/questions/852890/expectation-of-random-variables-ratio
%%% $L - w' \over (L - w') + (w - s)$
% $E(b/p) + 1 = E(b/p + p/p) = E((b + p)/p) = \kappa$
% $E(b/p) = \kappa-1$