The Open-Strategy Dictator Game: Cooperation Under Mutual Transparency

Michael Glass

cs.GTAug 14, 2026

The Open-Strategy Dictator Game: Cooperation Under Mutual Transparency

Authors: Michael Glass

Abstract

We introduce the Open-Strategy Dictator Game (OSDG), a variant of the classic dictator game in which each player's strategy is a natural-language document visible to all participants. The dictator's decision, to SHARE or TAKE an endowment, may depend on the text of the recipient's strategy. A large language model adjudicates each interaction by interpreting the dictator's strategy in the context of the recipient's. We run round-robin tournaments among diverse strategies and analyze the resulting payoff matrix using softmax equilibrium frequencies, dominance analysis, and sensitivity to the relative value of cooperation. Conditionally cooperative strategies, those that share with cooperators and take from exploiters, consistently dominate, while unconditional strategies (always share or always take) are weakly dominated. The results suggest that in environments where agents can inspect each other's decision procedures, conditional cooperation is evolutionarily robust across a wide range of payoff parameters.

Figures & tables

Strategy	Decision rule
Unconditional — the recipient’s strategy is ignored
Generous	Always share . The unconditional cooperator.
Selfish	Always take . The unconditional defector.
Trait tests — classify the recipient’s document
Intelligence	share with strategies whose decisions non-trivially depend on the recipient; take otherwise.
Anti-exploiter	share with strategies that appear cooperative or kind; take otherwise.

Table 1: The nine archetype strategies of Tournament 9, grouped by family in order of increasing sophistication of what the decision rule conditions on. The swatches give each strategy’s color in every figure. The full text of each document is reproduced in Appendix B .

Dictator	1	2	3	4	5	6	7	8	9
	Recipient
1. Mirror	S	S	S	S	S	S	S	S	T
2. Cooperation coalition	S	S	T	S	T	T	T	T	T
3. Chivalry	S	T	S	T	S	S	T	S	T
4. Conditional cooperator	S	S	S	S	S	S	S	T	T
5. Anti-exploiter	S	S	S	S	S	S	S	S	T

Table 2: Decision matrix for Tournament 9: the oracle’s decision when the row strategy dictates to the column strategy (S = share , T = take ). Strategies are numbered by final score; column numbers refer to the same ordering. Beyond the two unconditional rows, take decisions trace the second-order norms of Section 4.3 : Cooperation coalition takes from the unconditional cooperator and from every strategy that subsidizes defectors, while Chivalry takes from exactly the strategies that would take from an unconditional cooperator.

Strategy	1	Total	Dominated by
	Opponent
1. Mirror		59.055	—
2. Cooperation coalition		59.005	—
3. Chivalry		57.651	—
4. Conditional cooperator		56.298	2
5. Anti-exploiter		55.621	1, 3

Table 3: Payoff matrix

A

for Tournament 9 (Equation ( 3 )): the total payoff the row strategy earns against the column strategy across its dictator and recipient rounds; strategies are numbered by final score as in Table 2 . Only four values occur, shown light to dark in increasing order:

\ln 31\approx 3.43

(shares, not shared with);

\ln 61\approx 4.11

(takes, not shared with);

2\ln 31\approx 6.87

(mutual share);

\ln 61+\ln 31\approx 7.55

(takes while being shared with). Row sums give the leaderboard. Because darker is always better, weak dominance (Section 6 ) is visible directly: strategy

i

weakly dominates strategy

k

if row

i

is nowhere lighter than row

k

; the last column lists the strategies that weakly dominate each row. Six of the nine strategies are weakly dominated; the undominated set is exactly the top three — Mirror , Cooperation coalition , and Chivalry . Mirror beats Cooperation coalition by

0.050\approx g-5\delta

: the coalition harvests five subsidizers at

\delta

each but loses the single share

g

withheld by Chivalry , the one strategy in the pool that retaliates.

Figure 1: Strategy frequencies along the fixed-point iteration of Equation ( 10 ) (top two rows) and the author-type mixture model of Section 8 (bottom row; type masses

0.4

0.4

0.2

with

\lambda=0

\tfrac{1}{2}

1

). Each panel starts at iteration

0

from its initial distribution and stops at the first iterate within

0.01

(

L_{\infty}

) of the fixed point. Every panel starts from the uniform distribution except softmax,

\beta=20

(basin 2) , which starts from a sampled distribution inside the protection basin (Section 7.4 ). Hue encodes strategy family: greys are the unconditional strategies (light = Generous , dark = Selfish ), warm hues are the two second-order norms of Section 4.3 ( Cooperation coalition orange, Chivalry gold), blues are the direct reciprocators ( Mirror , Conditional cooperator light blue), and Universalizability (green), Intelligence (violet), and Anti-exploiter (pink) stand alone.

Figure 2: Basin of attraction at

\beta=20

for

2{,}000

initial distributions drawn uniformly from the simplex, projected onto the initial shares of the two second-order norms. Blue points converge to the enforcement equilibrium (basin 1,

79.6\%

of starts), orange points to the protection equilibrium (basin 2,

20.4\%

). The dashed line is the fitted logistic boundary; these two coordinates alone predict the outcome for

91\%

of starts (all nine coordinates:

94\%

Figure 3: Softmax equilibrium as a function of the share payoff

\sigma

( take normalized to

1

), at precision matched to the logarithmic-utility analysis (

\beta^{\prime}=20\ln 61

). The bands give the composition of the equilibrium reached from the uniform prior; colors as in Figure 1 . The dashed line is the fraction of

300

random starts that reach this equilibrium: where it falls below one, a rival basin exists. Dotted verticals mark the exchange-rate thresholds

\rho=1

(

\sigma=0.5

), below which a share costs more than it delivers and the field collapses to defection, and

\rho=4

(

\sigma=0.8

), above which the protection configuration becomes self-sustaining; the mark at

\sigma=\ln 31/\ln 61\approx 0.835

is the game of the preceding sections.

Game Theory

Game theory studies how interacting decision-makers choose strategies, form agreements, and shape outcomes under competition and cooperation.

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