Friday, June 24, 2016
Wilsonian Quantifier Trilemma: frogs and princes
In an earlier blog I referred to the “Wilsonian quantifiers”; ‘some but not all’ and ‘all or none’. These are to the ‘all’ and ‘exists’ quantifiers as ‘exclusive or’ and ‘if and only if’ are to ‘and’ and ‘or’. I have found a troika involving them.
Moe: No frogs are princes.
Larry: Some but not all frogs are princes.
Curly: All frogs are princes.
Moe, Larry and Curly all agree that frogs exist.
By 2/3 majorities each:
LK: Some frogs are princes.
ML: Some frogs are not princes.
KM: All or no frogs are princes.
The last can be read, “All frogs are equally princes.”
Thursday, June 23, 2016
Wednesday, June 22, 2016
Consider these four “non-transitive” dice:
Red: 4 4 4 4 4 4 Y
Yellow: 3 3 3 3 7 7 < <
Green: 2 2 2 6 6 6 G R
Blue: 1 1 5 5 5 5 > >
When you roll two of these dice:
Red loses to Blue 2/3 of the time;
Blue loses to Green 2/3 of the time;
Green loses to Yellow 2/3 of the time;
Yellow loses to Red 2/3 of the time.
A dominance loop! Also:
Red loses to Green 1/2 of the time;
Blue loses to Yellow 5/9 of the time;
These dice are fair, yet they are rigged. If two players choose dice to roll, then the second to choose has a 2:1 advantage. Whatever color your opponent picks, you should pick the next one on this cycle;
Blue → Green → Yellow → Red → Blue
This is the ‘redshift’ strategy; choose a lower-frequency color (except for the wrap-around red → blue). If you use this color code:
Blue = Winter; Green = Spring; Yellow = Summer; Red = Fall;
then the mnemonic is “next season”.
Here are some games based upon magic dice:
Double-Cross, Double Roll, Double Bet, White Magic, Black Magic and Black Double-Cross.
In the game Double-Cross, one player has the red and green dice, and the other player has the blue and yellow dice. They roll one die each, simultaneously. That way either player can undercut the other. This game combines choice with rigged chance.
In the game Double-Roll, one player is the first to choose a die for a round of several rolls, but the second player must twice as often. For a round of 9 rolls, a score of 3 to 6 is a draw. In the gambling game Double-Bet, the second player must bet twice as much. These schemes counter-bias the dice.
In the game White Magic, the winner of a roll must choose first the next time. White Magic’s winners tend to lose and its losers tend to win. White Magic smooths out all differences. In the long run it divides the pot evenly amongst the players. White Magic is very communal.
“Phooey,” say all real gamblers; they’d prefer White Magic’s dark shadow, Black Magic. In Black Magic, the loser must choose first on the next roll. Black Magic’s losers tend to keep losing, and its winners tend to keep winning. Black Magic accentuates random differences. In the long run it assigns a big winner. Black Magic is very competitive.
You can combine these games and get hybrids, like Black Double Cross, which combines Double-Cross and Black Magic. In this game, one player has red-green, the other has blue-yellow; whoever loses a roll must roll first next roll.
Care for a few rounds?