### Author Topic: Game Theory...?  (Read 443 times)

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#### heatmap

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##### Game Theory...?
« on: February 11, 2018, 05:25:10 AM »
This is a very hypothetical question on my part :

What if the wheels are using game theory to try to predict your every move, and 95% of the time does no matter what?

This is a very real description of what can be done to mathematically describe on that:

http://www.columbia.edu/~rs328/NashEquilibrium.pdf

Anyone ever hear of the movie: A beautiful mind?

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##### Re: Game Theory...?
« Reply #1 on: February 11, 2018, 05:35:24 AM »
I dont think you guys know how big this is.

#### Reyth

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##### Re: Game Theory...?
« Reply #2 on: February 11, 2018, 05:36:38 AM »
If the results are not random then I would not play.  Random is our friend.  AP may disagree, which is fine if you are playing AP.

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##### Re: Game Theory...?
« Reply #3 on: February 11, 2018, 05:47:27 AM »
screw it im posting the entire thing here

NASH EQUILIBRIUM
Nash equilibrium is a fundamental concept in the theory
of games and the most widely used method of predicting
the outcome of a strategic interaction in the social sciences.
A game (in strategic or normal form) consists of the
following three elements: a set of players, a set of actions
(or pure-strategies) available to each player, and a payoff
(or utility) function for each player. The payoff functions
represent each player’s preferences over action profiles,
where an action profile is simply a list of actions, one for
each player. A pure-strategy Nash equilibrium is an action
profile with the property that no single player can obtain
a higher payoff by deviating unilaterally from this profile.
This concept can best be understood by looking at
some examples. Consider first a game involving two players,
each of whom has two available actions, which we call
A and B. If the players choose different actions, they each
get a payoff of 0. If they both choose A, they each get 2,
and if they both choose B, they each get 1. This “coordination”
game may be represented as follows, where player
1 chooses a row, player 2 chooses a column, and the resulting
payoffs are listed in parentheses, with the first component
corresponding to player 1’s payoff:
The action profile (B,B) is an equilibrium, since a
unilateral deviation to A by any one player would result in
a lower payoff for the deviating player. Similarly, the
action profile (A,A) is also an equilibrium.
As another example, consider the game “matching
pennies,” which again involves two players, each with two
actions. Each player can choose either heads (H) or tails
(T); player 1 wins a dollar from player 2 if their choices
are the same, and loses a dollar to player 2 if they are not.
This game has no pure-strategy Nash equilibria.
In some cases, instead of simply choosing an action,
players may be able to choose probability distributions
over the set of actions available to them. Such randomizations
over the set of actions are referred to as mixed strategies.
Any profile of mixed strategies induces a probability
distribution over action profiles in the game. Under certain
assumptions, a player’s preferences over all such lotteries
can be represented by a function (called a von
Neumann-Morgenstern utility function) that assigns a real
number to each action profile. One lottery is preferred to
another if and only if it results in a higher expected value
of this utility function, or expected utility. A mixed strategy
Nash-equilibrium is then a mixed strategy profile with
the property that no single player can obtain a higher
value of expected utility by deviating unilaterally from this
profile.
The American mathematician John Nash (1950)
showed that every game in which the set of actions available
to each player is finite has at least one mixed-strategy
equilibrium. In the matching pennies game, there is a
mixed-strategy equilibrium in which each player chooses
heads with probability 1/2. Similarly, in the coordination
game of the above example, there is a third equilibrium in
which each player chooses action A with probability 1/3
and B with probability 2/3. Such multiplicity of equilibria
arises in many economically important games, and has
prompted a large literature on equilibrium refinements
with the purpose of identifying criteria on the basis of
which a single equilibrium might be selected.
Nash equilibria can sometimes correspond to outcomes
that are inefficient, in the sense that there exist
alternative outcomes that are both feasible and preferred
by all players. This is the case, for instance, with the equilibrium
(B,B) in the coordination game above. An even
more striking example arises in the prisoner’s dilemma
game, in which each player can either “cooperate” or
“defect,” and payoffs are as follows:
The unique Nash equilibrium is mutual defection, an
outcome that is worse for both players than mutual cooperation.
Now consider the game that involves a repetition
of the prisoner’s dilemma for n periods, where n is commonly
known to the two players. A pure strategy in this
repeated game is a plan that prescribes which action is to
be taken at each stage, contingent on every possible history
of the game to that point. Clearly the set of pure
strategies is very large. Nevertheless, all Nash equilibria of
this finitely repeated game involve defection at every
stage. When the number of stages n is large, equilibrium
payoffs lie far below the payoffs that could have been
attained under mutual cooperation.
It has sometimes been argued that the Nash prediction
in the finitely repeated prisoner’s dilemma (and in
many other environments) is counterintuitive and at odds
with experimental evidence. However, experimental tests
of the equilibrium hypothesis are typically conducted
with monetary payoffs, which need not reflect the preferences
of subjects over action profiles. In other words, individual
preferences over the distribution of monetary
payoffs may not be exclusively self-interested.
Furthermore, the equilibrium prediction relies on the
hypothesis that these preferences are commonly known to
all subjects, which is also unlikely to hold in practice.
To address this latter concern, the concept of Nash
equilibrium has been generalized to allow for situations in
which players are faced with incomplete information. If
each player is drawn from some set of types, such that the
probability distribution governing the likelihood of each
type is itself commonly known to all players, then we have
a Bayesian game. A pure strategy in this game is a function
that associates with each type a particular action. A BayesNash
equilibrium is then a strategy profile such that no
player can obtain greater expected utility by deviating to a
different strategy, given his or her beliefs about the distribution
of types from which other players are drawn.
Allowing for incomplete information can have dramatic
effects on the predictions of the Nash equilibrium
concept. Consider, for example, the finitely repeated prisoner’s
dilemma, and suppose that each player believes that
there is some possibility, perhaps very small, that his or her
opponent will cooperate in all periods provided that no
defection has yet been observed, and defect otherwise. If
the number of stages n is sufficiently large, it can be
shown that mutual defection in all stages is inconsistent
with equilibrium behavior, and that, in a well-defined
sense, the players will cooperate in most periods. Hence,
in applying the concept of Nash equilibrium to practical
situations, it is important to pay close attention to the
information that individuals have about the preferences,
beliefs, and rationality of those with whom they are strategically
interacting.
Nash Equilibrium
Figure 3
D (3,0) (1,1)
C (2,2) (0,3)
C D
INTERNATIONAL ENCYCLOPEDIA OF THE SOCIAL SCIENCES, 2ND EDITION 541
iess_B3_H-O 4/12/07 4:13 PM Page 541
Noncooperative Games; Prisoner’s Dilemma
(Economics)
BIBLIOGRAPHY
Cournot, A. A. 1838. Recherches sur les principes mathématiques
de la théorie des richesses. Paris: L. Hachette.
Fudenberg, Drew, and Jean Tirole. 1991. Game Theory.
Cambridge, MA: MIT Press.
Harsanyi, John C. 1967–1968. Games with Incomplete
Information Played by Bayesian Players. Management Science
14 (3): 159–182, 320–334, 486–502.
Harsanyi, John C., and Reinhard Selten. 1998. A General Theory
of Equilibrium Selection in Games. Cambridge, MA: MIT
Press.
Kreps, David, Paul Milgrom, John Roberts, and Robert Wilson.
1982. Rational Cooperation in the Finitely Repeated
Prisoner’s Dilemma. Journal of Economic Theory 27: 245–252.
Nash, John F. 1950. Equilibrium Points in N-Person Games.
Proceedings of the National Academy of Sciences 36 (1): 48–49.
Osborne, Martin J., and Ariel Rubinstein. 1994. A Course in
Game Theory. Cambridge, MA: MIT Press.
von Neumann, John, and Oskar Morgenstern. 1944. Theory of
Games and Economic Behavior. Princeton, NJ: Princeton
University Press.
Rajiv Sethi

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##### Re: Game Theory...?
« Reply #4 on: February 11, 2018, 11:12:02 AM »
ANNNND i was right

check this patent out

"Casino games"
US 20110105206

From the patent which is interesting :

a virtual player, instantiated by the game controller, configured to make the decisions requested by the multiplayer game engine during the play of the game, said virtual player comprising,
a decision engine configured to make the decisions requested by the multi-player game engine based upon at least a current position of the virtual player in the game, said decision engine based upon a game theory analysis of the game such that the decision engine implements an optimal strategy determined from the game theory analysis;
a behavioral engine configured to determine when to a trigger a behavior of the virtual player wherein the behavior when correctly interpreted by the live player increases the live player's chance of winning the game played against the virtual player; and
a personality engine configured to receive, when the behavior is triggered, information regarding the behavior, determine an action of a virtual character that is generated in response to the behavior; and generate a presentation of the virtual character performing the determined action;

#### vitorwally

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##### Re: Game Theory...?
« Reply #5 on: February 11, 2018, 11:38:49 AM »

In the past I had moments I thanked Von Neumann for creating the Princeton architecture which help us reach our capabilities today. If his knowledge

"...a function (called a von Neumann-Morgenstern utility function) that assigns a real number to each action profile..."

also plays a role on the wizardry you presented I'm gonna ask the admin of life to take off all the thanks I gave.
Aside, it seems to me that if they're using a concept from the ones you presented it's more likely to be the BayesNash equilibrium. Nonetheless, and if I read it correctly, the amount and quality of the information defines the effectiveness of the model. Both information the player has and information the model has of the context and the players.
This relates in some degree with the narrative I consistently hear at the airballs of my local casino. It's common to hear losing folks saying sentences like "What do you expected? The machine knows all the bets that were placed on the screens. It just has to induce the ball to a favorable spot for the casino".

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##### Re: Game Theory...?
« Reply #6 on: February 11, 2018, 11:47:40 AM »
@victorwally

the better post is my other one labeled "quantum gambling"

https://forum.roulette30.com/index.php?topic=2148.msg31456#msg31456

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##### Re: Game Theory...?
« Reply #7 on: February 11, 2018, 01:03:57 PM »
@vitorwally I mangled your username sorry about that. Do you think there is a way if the algorithm is known such as this that someone can develop a "way to work with it" somehow?

#### vitorwally

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##### Re: Game Theory...?
« Reply #8 on: February 11, 2018, 02:21:25 PM »
What I understood, based by both topics, is that the Nash Equilibrium is mainly used to create fair situations. In the casino point of view, create more "fair" overall payouts I guess? The difficult thing is that to work a plan against it you would probably need to know all other bet selections from the different players at the table. Wheels that could hypothetically use a Nash Equilibrium notion in their system are most likely the airballs.

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##### Re: Game Theory...?
« Reply #9 on: February 11, 2018, 02:38:39 PM »
In the example it said something like

Make sure Alice always gets her cut while bobs cut is a variable

I feel like I'm barking up the right tree here and this could get exciting if someone with cray maths skills can chime in here and settle this it would be great