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Exploratory experiment: Skeptical desire in multi-agent games
Experiment: exp_nash_equilibrium.sx | Type: exploratory runs on three small games
What was run
Gradient dynamics with the cosine-scaled projection are run on a 2-player zero-sum game, a 3-player coordination game, and the Prisoner's Dilemma. In the Prisoner's Dilemma three variants of the update are compared: no regularisation, a standard desire term, and a "skeptical desire" term that penalises strategies close to mutual defection when the welfare gap is large. The tables record the final strategies, welfare and convergence behaviour in each run. These are exploratory runs on small games; the numbers are the outcomes of the runs and nothing further is claimed for them.
Method
- Zero-sum game: 2-player matching pennies. Payoff matrix \(\begin{pmatrix}1 & -1 \\ -1 & 1\end{pmatrix}\). Run gradient dynamics with cosine-scaled projection for 1000 steps.
- 3-player coordination: Each player chooses strategy \(s_i \in [0, 1]\). Payoff is \(u_i = -|s_i - \bar{s}| + \epsilon_i\) where \(\bar{s}\) is the mean. Run with and without interaction matrix learning.
- Prisoner's Dilemma: Standard payoff matrix (T=5, R=3, P=1, S=0). Run three variants:
- No regularisation (baseline)
- Standard desire regularisation
- Skeptical desire regularisation
- Measure: final strategy, social welfare, fraction of the welfare gap to mutual cooperation, convergence time.
Results
Zero-Sum Game
| Method | Player 1 Strategy | Player 2 Strategy | Exploitability | Steps to settle |
|---|
| Gradient dynamics | 0.50 ± 0.12 | 0.50 ± 0.11 | 0.24 | Did not settle |
| With cosine projection | 0.500 ± 0.003 | 0.500 ± 0.003 | 0.006 | 142 |
With the projection the run settles at the mixed strategy (0.5, 0.5), which is the game's Nash point; plain gradient dynamics oscillates around it.
3-Player Coordination
| Method | Strategy Variance | Mean Payoff | Steps to settle |
|---|
| Independent | 0.082 | 0.71 | 340 |
| Interaction matrix | 0.011 | 0.94 | 87 |
Prisoner's Dilemma
| Method | P(Cooperate) | Social Welfare | Fraction of gap closed | Mutual-defection welfare |
|---|
| No regularisation | 0.00 | 2.00 | 0.0% | 2.00 |
| Standard desire | 0.41 | 3.28 | 52.1% | 2.00 |
| Skeptical desire | 0.72 | 5.34 | 83.5% | 2.00 |
Mutual-cooperation welfare = 6.0; mutual-defection welfare = 2.0. The skeptical-desire run ends at welfare 5.34, 83.5% of the way from 2.0 to 6.0.
Convergence Dynamics
| Method | Steps to settle | Oscillation Amplitude | Final \(S\) |
|---|
| No regularisation | 45 | 0.000 | 1.000 |
| Standard desire | 280 | 0.031 | 0.871 |
| Skeptical desire | 410 | 0.008 | 0.934 |
Observations
- In matching pennies the projected dynamics settle at the mixed strategy (0.5, 0.5) within 142 steps; unprojected gradient dynamics does not settle within 1000 steps.
- In the 3-player coordination game, learning the interaction matrix reduces strategy variance from 0.082 to 0.011 and the settling time from 340 to 87 steps in this run.
- In the Prisoner's Dilemma, the skeptical-desire term adds a penalty on strategies near mutual defection when the welfare gap is large; the run ends at P(Cooperate) = 0.72 and welfare 5.34. The unregularised run ends at mutual defection (welfare 2.0).
- The skeptical-desire run does not reach mutual cooperation; the temptation payoff still pulls toward defection.
Reproducibility
../simplex/build/sxc exp_nash_equilibrium.sx -o build/exp_nash_equilibrium.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_nash_equilibrium.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_nash_equilibrium \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_nash_equilibrium
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