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This document is page 21 of an academic paper or scientific report discussing game theory, specifically 'Learning Dynamics of the Envelope Game.' The text defines strategies labeled CWOL, CWL, and ALLD, and discusses mathematical equilibrium conditions involving variables like temptation, probability, and cooperation. It describes running simulations (1000 time series) using replicator dynamics to model strategy evolution. The page bears a 'HOUSE_OVERSIGHT_026541' stamp, identifying it as part of a document dump related to a House Oversight Committee investigation, likely concerning Jeffrey Epstein's funding of academic research (e.g., evolutionary dynamics).
Organizations (1)
| Name | Type | Context |
|---|---|---|
| House Oversight Committee |
Document bears the stamp 'HOUSE_OVERSIGHT_026541', indicating it was part of a document production for a congressiona...
|
Key Quotes (4)
"ALLD is always an equilibrium of the envelope game."Source
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Quote #1
"CWOL is an equilibrium if a/(1 - w) > c_l p + c_h (1 - p)."Source
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Quote #2
"We apply the replicator dynamic to the envelope game restricted to the strategies represented in figure 2."Source
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Quote #3
"The replicator dynamic also models learning dynamics such as reinforcement learning or prestige-biased imitation."Source
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Quote #4
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Complete text extracted from the document (2,095 characters)
same cell; for details of calculations leading to payoffs, see appendix). This calculation
is the result of the following logic: each time the temptation is low, player 1 cooperates
and gets a, player 2 gets b, and the game continues with probability w until the first
time the temptation is high. We refer to the strategy where player 1 cooperates without
looking (top row) as CWOL. We also refer to the strategy pair where player 1 CWOLs
and player 2 continues if player 1 CWOLs (first column) as CWOL. We refer to the
strategy pairs where player 1 cooperates with or without looking and player 2 continues
if player 1 cooperates (first and second row, and middle middle column) as CWL. We
refer to the strategy pair where player 1 always defects and player 2 always exits (bottom
row and rightmost column) as ALLD. ALLD is always an equilibrium of the envelope
game. CWOL is an equilibrium if a/(1 - w) > c_l p + c_h (1 - p). CWL is an equilibrium if
a/(1 - w) > c_h. This region is a subset of the region for which CWOL is an equilibrium.
is the result of the following logic: each time the temptation is low, player 1 cooperates
and gets a, player 2 gets b, and the game continues with probability w until the first
time the temptation is high. We refer to the strategy where player 1 cooperates without
looking (top row) as CWOL. We also refer to the strategy pair where player 1 CWOLs
and player 2 continues if player 1 CWOLs (first column) as CWOL. We refer to the
strategy pairs where player 1 cooperates with or without looking and player 2 continues
if player 1 cooperates (first and second row, and middle middle column) as CWL. We
refer to the strategy pair where player 1 always defects and player 2 always exits (bottom
row and rightmost column) as ALLD. ALLD is always an equilibrium of the envelope
game. CWOL is an equilibrium if a/(1 - w) > c_l p + c_h (1 - p). CWL is an equilibrium if
a/(1 - w) > c_h. This region is a subset of the region for which CWOL is an equilibrium.
Figure 3: Learning Dynamics of the Envelope Game
We apply the replicator dynamic to the envelope game restricted to the strategies repre-
sented in figure 2. The replicator dynamic describes strategies evolving over time under
the assumption that the rate of reproduction within each population is proportional to
the fitness relative to that type's other strategies. The replicator dynamic also models
learning dynamics such as reinforcement learning or prestige-biased imitation. We run
1000 time series with randomly seeded strategy frequencies for a range of values of a, and
record the frequency with which they stabilize in one of the strategy pairs identified in
figure 2, or in a behaviorally equivalent equilibrium, as presented in the simplexes. We
vary the value of a along the x-axis. The y-axis represents frequencies, and each colored
line presents the frequency of the strategy pair. The parameter region where the strategy
pair is supported as an equilibrium is shaded in light red. CWOL is supported as an
sented in figure 2. The replicator dynamic describes strategies evolving over time under
the assumption that the rate of reproduction within each population is proportional to
the fitness relative to that type's other strategies. The replicator dynamic also models
learning dynamics such as reinforcement learning or prestige-biased imitation. We run
1000 time series with randomly seeded strategy frequencies for a range of values of a, and
record the frequency with which they stabilize in one of the strategy pairs identified in
figure 2, or in a behaviorally equivalent equilibrium, as presented in the simplexes. We
vary the value of a along the x-axis. The y-axis represents frequencies, and each colored
line presents the frequency of the strategy pair. The parameter region where the strategy
pair is supported as an equilibrium is shaded in light red. CWOL is supported as an
21
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