SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figure 2: Left: configuration 1 (used for RS212, RS213). Right: configuration 2 (used for RS222, RS223). In both cases the position of the two drivers is randomly chosen at the beginning of the game, edge costs are unitary, and the reward for each node is indicated between curly brackets.
G Additional experimental results
G.1 Investigation of lower equilibrium approximation
In Table 2 we show results, using the same format as Table 1 shown in the body, for the case in which the approximation  $\varepsilon$  is set to be  $0.1\%$  of the payoff range of the game, as opposed to the  $1\%$  threshold of the body.
Table 2: Comparison between the linear-programming-based algorithm ('LP') of Zhang and Sandholm [100] and our learning-based approach ('Ours'), for the problem of computing an approximate optimal equilibrium within tolerance  $\varepsilon$  set to  ${0.1}\%$  of the payoff range of the game.
Game	# Nodes	NFCCE	EFCCE	EFCE	COMM	CERT
LP	Ours	LP	Ours	LP	Ours	LP	Ours	LP	Ours
B2222	1573	0.00s	0.00s	0.00s	0.02s	0.00s	0.03s	3.00s	1m 5s	0.00s	0.04s
B2322	23839	1.00s	0.02s	3.00s	1.42s	9.00s	4.11s	timeout	17m 30s	2.00s	2.82s
B2323	254239	6.00s	0.66s	1m 29s	30.04s	3m 40s	1m 28s	timeout	timeout	39.00s	1m 24s
B2324	1420639	41.00s	5.25s	timeout	5m 49s	timeout	timeout	timeout	timeout	timeout	timeout
D32	1017	0.00s	0.03s	0.00s	0.04s	14.00s	0.92s	1.00s	0.26s	0.00s	0.03s
D33	27622	3m 22s	44.41s	timeout	10m 27s	timeout	timeout	timeout	16m 38s	6.00s	6.87s
GL3	7735	0.00s	0.06s	1.00s	0.07s	0.00s	0.06s	timeout	36.83s	0.00s	0.11s
K35	1501	55.00s	2.46s	53.00s	3.05s	1m 5s	2.99s	1.00s	0.09s	0.00s	0.02s
L3132	8917	28.00s	2.13s	11m 26s	22.14s	9m 41s	26.68s	13.00s	15.41s	1.00s	0.62s
L3133	12688	45.00s	2.83s	timeout	35.86s	26m 52s	22.31s	17.00s	15.27s	1.00s	1.25s
L3151	19981	timeout	54.66s	timeout	timeout	timeout	timeout	timeout	1m 15s	2.00s	0.91s
L3223	15659	5.00s	1.73s	1m 21s	8.58s	2m 38s	20.44s	26.00s	1m 43s	1.00s	2.00s
L3523	1299005	timeout	4m 4s	timeout	timeout	timeout	timeout	timeout	timeout	timeout	timeout
S2122	705	0.00s	0.00s	0.00s	0.02s	0.00s	0.07s	3.00s	6.14s	0.00s	0.03s
S2123	4269	0.00s	0.02s	1.00s	0.14s	1.00s	0.37s	1m 51s	10m 8s	1.00s	0.41s
S2133	9648	1.00s	0.05s	3.00s	0.17s	4.00s	0.95s	timeout	timeout	3.00s	1.99s
S2254	712552	2m 0s	32.14s	timeout	42.65s	timeout	9m 2s	timeout	timeout	timeout	6m 50s
S2264	1303177	3m 48s	57.76s	timeout	1m 16s	timeout	timeout	timeout	timeout	timeout	timeout
TP3	910737	1m 43s	14.28s	timeout	20.81s	timeout	26.28s	timeout	timeout	timeout	52.76s
RS212	598	0.00s	0.00s	0.00s	0.00s	0.00s	0.00s	2.00s	0.02s	0.00s	0.00s
RS222	734	0.00s	0.01s	0.00s	0.01s	0.00s	0.02s	3.00s	0.03s	0.00s	0.00s
RS213	6274	timeout	43.46s	timeout	45.00s	timeout	2m 28s	7m 30s	27.19s	0.00s	0.03s
RS223	6238	timeout	timeout	timeout	timeout	timeout	timeout	9m 16s	15.68s	1.00s	0.05s
We observe that none of the results change qualitatively when this increased precision is considered.
G.2 Game values and size of Mediator's strategy space
Tables 3 and 4 reports optimal equilibrium values for all games and all equilibrium concepts for which the LP algorithm was able to compute an exact value (We restrict to the cases solvable by LP because the Lagrangian relaxations only compute an  $\varepsilon$ -equilibrium, but the mediator objective in an  $\varepsilon$ -equilibrium could be arbitrarily far away from the mediator objective in an exact equilibrium). We hope that these will be good references for future researchers interested in this topic. Table 5 reports the size of the strategy space of the mediator player in the two-player zero-sum game that captures the computation of optimal equilibria, in terms of the number of decision points and edges. For correlated notions, this number may be exponential in the original game size; for communication and certification notions, it will always be polynomial.
Table 3: Optimal equilibrium value for correlated equilibrium concepts. 'Pl. 1' is the utility for Player 1 in the Player 1-optimal equilibrium. 'Pl. 2' and 'Pl. 3' are similar. In two-player games, 'SW' is the welfare of the welfare-maximizing equilibrium. (these three values, of course, may come from three different equilibria.) The three-player games are zero-sum, so optimizing welfare makes no sense (the welfare is always zero).
Game	NFCCE	EFCCE	EFCE
Pl. 1	Pl. 2	SW	Pl. 1	Pl. 2	SW	Pl. 1	Pl. 2	SW
B2222	0.281	0.094	0.000	-0.027	-0.338	-0.525	-0.031	-0.338	-0.525
B2322	0.181	0.097	0.000	-0.043	-0.123	-0.317	-0.045	-0.123	-0.317
B2323	0.250	0.125	0.000	0.000	-0.125	-0.375	-0.001	-0.125	-0.375
B2324	0.306	0.139	0.000	—	—	—	—	—	—
S2122	11.636	5.999	13.636	7.652	5.043	9.565	7.262	3.841	9.078
S2123	11.636	5.999	13.636	8.000	5.191	10.000	8.000	4.611	10.000
S2133	15.182	6.992	18.182	12.000	6.557	15.000	12.000	6.407	15.000
S2254	23.571	12.830	28.571	—	—	—	—	—	—
S2264	27.333	13.840	33.333	—	—	—	—	—	—
U212	3.123	3.123	6.010	3.071	3.071	6.010	3.071	3.071	6.010
U213	—	—	—	—	—	—	—	—	—
U222	3.765	3.765	7.188	3.719	3.719	7.176	3.719	3.719	7.176
U223	—	—	—	—	—	—	—	—	—
Pl. 1	Pl. 2	Pl. 3	Pl. 1	Pl. 2	Pl. 3	Pl. 1	Pl. 2	Pl. 3
D32	0.250	0.250	0.131	0.250	0.250	0.000	0.250	0.250	0.000
D33	0.422	0.284	0.239	—	—	—	—	—	—
GL3	2.505	2.505	2.505	2.476	2.476	2.476	2.467	2.467	2.467
K35	-0.011	0.017	0.057	-0.016	0.015	0.052	-0.016	0.013	0.052
L3132	0.571	0.504	0.606	0.519	—	—	0.467	0.422	—
L3133	0.419	0.348	0.416	—	—	—	—	—	—
L3151	—	—	—	—	—	—	—	—	—
L3223	1.079	0.992	1.146	0.984	0.959	1.033	0.887	0.883	0.861
L3523	—	—	—	—	—	—	—	—	—
TP3	1.466	1.477	1.037	—	—	—	—	—	—
Table 4: Optimal equilibrium value for communication and certification equilibrium concepts. 'Pl. 1', 'Pl. 2', 'Pl. 3', and 'SW' have the same meaning as in the previous table.
Game	COMM	NFCCERT	CCERT	CERT
Pl. 1	Pl. 2	SW	Pl. 1	Pl. 2	SW	Pl. 1	Pl. 2	SW	Pl. 1	Pl. 2	SW
B2222	-0.187	-0.562	-0.750	0.281	0.094	0.000	-0.027	-0.338	-0.525	-0.027	-0.338	-0.525
B2322	—	—	—	0.181	0.097	0.000	-0.043	-0.123	-0.317	-0.043	-0.123	-0.317
B2323	—	—	—	0.250	0.125	0.000	0.000	-0.125	-0.375	0.000	-0.125	-0.375
B2324	—	—	—	0.306	0.139	0.000	—	—	—	—	—	—
S2122	0.820	0.000	0.820	50.000	8.508	50.000	8.000	5.191	10.000	8.000	4.611	10.000
S2123	0.820	0.000	0.820	50.000	8.508	50.000	8.000	5.191	10.000	8.000	4.611	10.000
S2133	—	—	—	50.000	8.671	50.000	12.000	6.557	15.000	12.000	6.407	15.000
S2254	—	—	—	100.000	17.284	100.000	20.000	12.190	25.000	—	—	—
S2264	—	—	—	100.000	17.442	100.000	—	—	—	—	—	—
U212	3.184	3.143	6.173	3.184	3.173	6.173	3.184	3.159	6.173	3.184	3.143	6.173
U213	5.160	5.171	9.592	5.316	5.429	9.622	5.204	5.298	9.622	5.196	5.276	9.622
U222	4.023	3.812	7.594	4.023	3.930	7.594	4.023	3.905	7.594	4.023	3.839	7.594
U223	6.537	6.326	11.464	6.867	6.783	11.516	6.631	6.582	11.513	6.576	6.398	11.485
Pl. 1	Pl. 2	Pl. 3	Pl. 1	Pl. 2	Pl. 3	Pl. 1	Pl. 2	Pl. 3	Pl. 1	Pl. 2	Pl. 3
D32	0.250	0.250	0.042	0.500	0.250	0.250	0.250	0.250	0.250	0.250	0.250	0.250
D33	—	—	—	0.580	0.296	0.284	0.444	0.296	0.284	0.432	0.296	0.272
GL3	—	—	—	2.505	2.505	2.505	2.505	2.505	2.505	2.467	2.468	2.468
K35	0.022	0.050	0.088	0.092	0.106	0.169	0.092	0.090	0.169	0.086	0.090	0.169
L3132	0.646	0.618	0.723	0.853	0.779	0.802	0.853	0.779	0.802	0.853	0.779	0.802
L3133	0.441	0.459	0.590	0.646	0.654	0.709	0.646	0.654	0.709	0.646	0.654	0.709
L3151	—	—	—	0.179	0.197	0.222	0.179	0.182	0.222	0.171	0.182	0.222
L3223	1.011	0.915	1.020	1.379	1.556	1.451	1.379	1.556	1.451	1.379	1.556	1.451
L3523	—	—	—	2.000	2.000	2.000	—	—	—	—	—	—
TP3	—	—	—	1.739	1.506	1.083	—	—	—	—	—	—
Table 5: Dimension of the mediator's decision space in terms of number of decision points ('Dec. pts.') and edges.
Game	NFCCE	EFCCE	EFCE	COMM	NFCCERT	CCERT	CERT
Dec. pts.	Edges	Dec. pts.	Edges	Dec. pts.	Edges	Dec. pts.	Edges	Dec. pts.	Edges	Dec. pts.	Edges	Dec. pts.	Edges
B2222	1429	6915	5001	21868	4212	20534	16341	65577	663	2739	1430	5854	3590	14638
B2322	11707	89519	66181	340619	67219	503145	681523	4090261	5661	34227	13940	84080	52640	317180
B2323	164707	1022286	1067881	5446015	1032019	7271972	—	—	77661	394227	244340	1236080	959840	4853180
B2324	1316707	6397418	8296681	41633816	8160018	4926467	—	—	596061	2467827	2188340	9012080	8476640	34920380
D32	3956	33823	5381	51593	53402	536485	12794	45070	472	1796	504	1844	2484	9176
D33	417625	1165451	1599919	71372690	—	—	2854524	10450812	11292	44382	18396	68937	135504	520665
GL3	8898	30021	10680	37041	5637	16950	182289	547086	2343	7104	3138	9474	5058	15234
K35	52277	3592121	60257	3826201	61217	4535281	9745	29235	1005	3015	1075	3225	2315	6945
L3132	131012	1222128	689890	15329595	694381	8400513	326730	980190	7773	23319	15789	47367	32055	96165
L3133	155297	1500087	1002685	26166405	1010749	14519676	365187	109561	9960	29880	21534	64602	44868	134604
L3151	1697120	34405970	—	—	—	—	1784965	5354895	18285	54855	36115	108345	72395	217185
L3223	91735	614847	405691	5617510	678365	4999142	1234394	400404	14186	46656	36298	118576	83786	273276
L3523	7595335	58635336	—	—	—	—	—	—	1115978	3887736	5617402	19357364	14863826	5121448
S2122	651	2903	2061	7847	1629	6227	4071	12629	408	1396	825	2627	1749	5465
S2123	4413	19049	17883	72377	13113	52559	146631	454565	2496	8452	6873	21959	15717	49349
S2133	9424	45931	40960	171915	38732	165859	778108	2425875	5112	18566	15000	49607	44084	139851
S2254	617056	3758737	3974008	18655297	5470186	25303237	—	—	327992	1300694	1332064	4615207	6089698	19348765
S2264	1103369	7284509	7291859	34837769	—	—	—	—	579182	2358134	2402695	8425369	12809119	40551631
TP3	2355864	7145312	3574464	11720048	2211712	6714256	—	—	1070544	3273072	1739488	5273184	3594352	10896416
RS212	658	15410	658	10604	538	9000	3317	14338	213	902	182	768	182	768
RS222	1625	55123	1625	35047	1495	32643	4142	15914	290	1098	252	952	252	952
RS213	61122	95194268	62704	95250604	97070	124453191	365621	1638786	2459	10870	2808	12416	4288	19160
RS223	—	—	—	—	—	—	299162	1184114	2778	10854	3224	12600	4500	17776
G.3 Detailed breakdown by equilibrium and objective function (two-player games)
For each two-player game, we try three different objective functions: maximizing the utility of Player 1, maximizing the utility of Player 2, and maximizing social welfare. For each objective, we stop the optimization at the approximation level defined as  $1\%$  of the payoff range of the game.
We use online optimistic gradient descent to update the Lagrange multipliers of each player (see Appendix E.1). For each objective, we report the following information:
- The runtime of the linear-programming-based algorithm ('LP') of Zhang and Sandholm [100].
- The runtime of our algorithm where each agent (player or mediator) uses the Discounted CFR ('DCFR') algorithm set up with the hyperparameters recommended in the work by Brown and Sandholm [12]. In the table, we report the best runtime across all choices of the stepsize hyperparameter  $\eta \in \{0.01, 0.1, 1.0, 10.0\}$  used in online optimistic gradient descent to update the Lagrange multipliers. The value of  $\eta$  that produces the reported runtime is noted in square brackets.
- The runtime of our algorithm where each agent (player or mediator) uses the Predictive  $\mathrm{CFR}^{+}$  ( $\mathrm{PCFR}^{+}$ ) algorithm of [33]. In the table, we report the best runtime across all choices of the stepsize hyperparameter  $\eta \in \{0.01, 0.1, 1.0, 10.0\}$  used in online optimistic gradient descent to update the Lagrange multipliers. The value of  $\eta$  that produces the reported runtime is again noted in square brackets.
G.3.1 Results for NFCCE solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)
B2222	0.00s	0.00s	[1.0]	0.00s	[0.1]	0.00s	0.00s	[1.0]	0.00s	[0.1]	0.00s	[1.0]
B2322	0.00s	0.03s	[0.1]	0.07s	[0.1]	0.00s	0.04s	[0.1]	0.04s	[1.0]	0.00s	[0.1]
B2323	7.00s	1.05s	[0.1]	1.61s	[0.1]	6.00s	1.01s	[0.1]	1.63s	[0.1]	6.00s	0.33s
B2324	50.00s	15.57s	[0.1]	20.01s	[0.1]	37.00s	12.80s	[0.1]	20.91s	[0.1]	38.00s	2.73s
S2122	0.00s	0.00s	[0.1]	0.00s	[0.1]	0.00s	0.00s	[0.1]	0.00s	[0.1]	0.00s	[0.1]
S2123	0.00s	0.01s	[1.0]	0.01s	[0.1]	0.00s	0.01s	[0.1]	0.01s	[0.1]	0.00s	[0.1]
S2133	1.00s	0.02s	[1.0]	0.02s	[0.1]	1.00s	0.02s	[0.1]	0.03s	[0.1]	1.00s	0.02s
S2254	2m 1s	6.96s	[0.1]	11.43s	[0.1]	1m 14s	10.43s	[0.1]	17.72s	[0.1]	1m 58s	7.43s
S2264	3m 36s	13.96s	[0.1]	23.25s	[0.1]	2m 24s	18.46s	[0.1]	35.04s	[0.1]	3m 43s	11.74s
RS212	0.00s	0.00s	[1.0]	0.00s	[1.0]	0.00s	0.00s	[10.0]	0.00s	[1.0]	0.00s	[10.0]
RS222	0.00s	0.00s	[1.0]	0.01s	[0.1]	0.00s	0.00s	[1.0]	0.01s	[1.0]	0.00s	[0.01]
RS213	timeout	34.52s	[1.0]	1m 9s	[0.1]	timeout	20.29s	[1.0]	41.66s	[0.1]	timeout	14.68s
RS223	timeout	timeout	—	timeout	—	timeout	—	timeout	—	timeout	—	timeout
G.3.2 Results for EFCCE solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)
B2222	0.00s	0.01s [1.0]	0.02s [1.0]	0.00s	0.01s [1.0]	0.02s [1.0]	0.00s	0.01s [1.0]	0.01s [1.0]	0.01s [1.0]	0.01s [1.0]	0.01s [1.0]
B2322	3.00s	0.41s [1.0]	1.13s [1.0]	3.00s	0.69s [1.0]	1.19s [1.0]	3.00s	0.69s [1.0]	1.33s [1.0]	1.33s [1.0]	1.33s [1.0]	1.33s [1.0]
B2323	1m 35s	9.11s [1.0]	22.38s [1.0]	1m 30s	12.39s [1.0]	23.59s [1.0]	1m 21s	14.23s [1.0]	20.30s [1.0]	20.30s [1.0]	20.30s [1.0]	20.30s [1.0]
B2324	timeout	1m 53s [1.0]	3m 35s [1.0]	timeout	2m 44s [1.0]	4m 29s [1.0]	timeout	3m 1s [1.0]	4m 16s [1.0]	4m 16s [1.0]	4m 16s [1.0]	4m 16s [1.0]
S2122	0.00s	0.01s [1.0]	0.01s [1.0]	0.00s	0.00s [0.1]	0.01s [0.1]	0.00s	0.01s [1.0]	0.02s [1.0]	0.02s [1.0]	0.02s [1.0]	0.02s [1.0]
S2123	1.00s	0.03s [1.0]	0.19s [0.1]	1.00s	0.09s [0.1]	0.05s [0.1]	1.00s	0.06s [1.0]	0.16s [1.0]	0.16s [1.0]	0.16s [1.0]	0.16s [1.0]
S2133	3.00s	0.12s [1.0]	0.31s [1.0]	2.00s	0.20s [0.1]	0.22s [0.1]	3.00s	0.11s [1.0]	0.41s [1.0]	0.41s [1.0]	0.41s [1.0]	0.41s [1.0]
S2254	timeout	28.81s [1.0]	27.31s [0.1]	timeout	37.43s [0.1]	53.08s [0.1]	timeout	22.01s [1.0]	36.83s [0.1]	36.83s [0.1]	36.83s [0.1]	36.83s [0.1]
S2264	timeout	46.07s [0.1]	1m 24s [0.1]	timeout	2m 18s [0.1]	2m 26s [0.1]	timeout	39.23s [0.1]	1m 9s [0.1]	1m 9s [0.1]	1m 9s [0.1]	1m 9s [0.1]
RS212	0.00s	0.00s [1.0]	0.00s [1.0]	0.00s	0.00s [1.0]	0.00s [1.0]	0.00s	0.00s [0.1]	0.00s [0.1]	0.00s [0.1]	0.00s [0.1]	0.00s [0.1]
RS222	0.00s	0.00s [1.0]	0.01s [0.1]	0.00s	0.00s [1.0]	0.01s [1.0]	0.00s	0.00s [0.1]	0.00s [0.1]	0.00s [0.1]	0.00s [0.1]	0.00s [0.1]
RS213	timeout	28.80s [1.0]	1m 50s [1.0]	timeout	31.02s [1.0]	1m 11s [1.0]	timeout	15.54s [0.1]	37.73s [0.1]	37.73s [0.1]	37.73s [0.1]	37.73s [0.1]
RS223	timeout	timeout [—]	timeout [—]	timeout	timeout [—]	timeout [—]	timeout	timeout [—]	timeout [—]	timeout [—]	timeout [—]	—
G.3.3 Results for EFCE solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)
B2222	0.00s	0.01s	[1.0]	0.03s	[1.0]	0.00s	0.06s	[1.0]	0.05s	[0.1]	0.00s	0.03s
B2322	9.00s	1.23s	[1.0]	2.97s	[1.0]	9.00s	4.63s	[1.0]	4.68s	[1.0]	9.00s	1.60s
B2323	3m 54s	48.40s	[1.0]	1m 28s	[1.0]	4m 9s	1m 38s	[1.0]	1m 27s	[1.0]	3m 40s	45.12s
B2324	timeout	9m 3s	[1.0]	13m 8s	[1.0]	timeout	timeout	[—]	10m 21s	[1.0]	timeout	14m 30s
S2122	0.00s	0.01s	[0.1]	0.02s	[1.0]	0.00s	0.02s	[0.1]	0.04s	[0.1]	0.00s	0.02s
S2123	1.00s	0.09s	[1.0]	0.23s	[0.1]	1.00s	0.34s	[0.1]	0.43s	[1.0]	1.00s	0.15s
S2133	4.00s	0.52s	[1.0]	0.77s	[0.1]	3.00s	1.86s	[0.1]	1.31s	[0.1]	3.00s	0.49s
S2254	timeout	2m 17s	[0.1]	2m 10s	[0.1]	timeout	timeout	[—]	timeout	[—]	timeout	3m 34s
S2264	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
RS212	0.00s	0.00s	[1.0]	0.00s	[1.0]	0.00s	0.00s	[1.0]	0.00s	[1.0]	0.00s	0.00s
RS222	0.00s	0.00s	[1.0]	0.01s	[0.1]	0.00s	0.00s	[1.0]	0.01s	[1.0]	0.00s	0.00s
RS213	timeout	35.37s	[1.0]	1m 28s	[0.1]	timeout	32.49s	[1.0]	1m 27s	[1.0]	timeout	23.37s
RS223	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
G.3.4 Results for COMM solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)
B2222	2.00s	0.88s	[1.0]	1.14s	[1.0]	2.00s	1.23s	[10.0]	0.89s	[1.0]	2.00s	1.49s
B2322	timeout	5m 47s	[1.0]	10m 17s	[1.0]	timeout	3m 45s	[1.0]	5m 2s	[1.0]	timeout	4m 41s
B2323	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
B2324	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
S2122	2.00s	0.21s	[0.1]	0.36s	[0.1]	2.00s	0.48s	[0.01]	0.48s	[0.1]	2.00s	0.36s
S2123	1m 30s	38.95s	[0.1]	1m 7s	[0.1]	1m 36s	1m 10s	[0.01]	1m 52s	[0.01]	1m 33s	59.63s
S2133	timeout	7m 34s	[0.01]	4m 26s	[0.1]	timeout	7m 27s	[0.01]	14m 12s	[0.01]	timeout	12m 11s
S2254	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
S2264	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
RS212	2.00s	0.01s	[1.0]	0.01s	[1.0]	2.00s	0.01s	[1.0]	0.03s	[10.0]	2.00s	0.01s
RS222	3.00s	0.01s	[1.0]	0.01s	[0.1]	3.00s	0.02s	[10.0]	0.04s	[1.0]	3.00s	0.02s
RS213	6m 51s	11.05s	[1.0]	10.24s	[1.0]	6m 27s	14.66s	[10.0]	12.83s	[1.0]	6m 25s	9.00s
RS223	8m 41s	5.79s	[10.0]	10.49s	[1.0]	8m 24s	8.45s	[1.0]	11.14s	[1.0]	8m 54s	4.00s
G.3.5 Results for NFCCERT solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFFR)	Ours (PCFR+)	LP	Ours (DCFFR)	Ours (PCFR+)	LP	Ours (DCFFR)	Ours (PCFR+)
B2222	0.00s	0.00s	[1.0]	0.00s	[0.1]	0.00s	0.00s	[1.0]	0.00s	[0.1]	0.00s	0.00s	[0.01]	0.00s	[10.0]
B2322	0.00s	0.02s	[1.0]	0.02s	[0.1]	0.00s	0.02s	[0.1]	0.02s	[0.1]	0.00s	0.01s	[0.01]	0.01s	[0.1]
B2323	2.00s	0.62s	[0.1]	0.63s	[0.1]	1.00s	0.48s	[0.1]	0.73s	[0.1]	2.00s	0.14s	[1.0]	0.18s	[0.1]
B2324	11.00s	4.88s	[0.1]	11.06s	[0.1]	11.00s	5.24s	[0.1]	9.92s	[0.1]	10.00s	1.82s	[0.01]	2.51s	[1.0]
S2122	0.00s	0.00s	[0.01]	0.00s	[0.01]	0.00s	0.00s	[0.01]	0.00s	[0.1]	0.00s	0.00s	[1.0]	0.00s	[0.01]
S2123	0.00s	0.00s	[0.01]	0.00s	[0.01]	0.00s	0.01s	[0.01]	0.00s	[0.1]	0.00s	0.00s	[0.01]	0.00s	[0.01]
S2133	0.00s	0.01s	[0.01]	0.01s	[0.1]	0.00s	0.01s	[1.0]	0.01s	[0.1]	0.00s	0.01s	[0.1]	0.01s	[0.1]
S2254	25.00s	1.23s	[0.01]	2.49s	[0.01]	24.00s	3.02s	[0.1]	3.02s	[0.1]	28.00s	1.32s	[0.1]	2.00s	[0.01]
S2264	56.00s	2.71s	[0.1]	4.93s	[0.01]	42.00s	5.43s	[0.1]	6.88s	[0.01]	50.00s	2.73s	[0.1]	3.65s	[0.01]
RS212	0.00s	0.00s	[0.1]	0.00s	[10.0]	0.00s	0.00s	[10.0]	0.00s	[0.1]	0.00s	0.00s	[0.1]	0.00s	[0.01]
RS222	0.00s	0.00s	[10.0]	0.00s	[1.0]	0.00s	0.00s	[10.0]	0.00s	[1.0]	0.00s	0.00s	[1.0]	0.00s	[0.1]
RS213	0.00s	0.00s	[0.1]	0.00s	[0.01]	0.00s	0.00s	[10.0]	0.00s	[0.1]	0.00s	0.00s	[1.0]	0.00s	[0.01]
RS223	0.00s	0.00s	[1.0]	0.00s	[0.1]	0.00s	0.00s	[10.0]	0.01s	[1.0]	0.00s	0.00s	[0.01]	0.01s	[1.0]
G.3.6 Results for CCERT solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFFR)	Ours (PCFR+)	LP	Ours (DCFFR)	Ours (PCFR+)	LP	Ours (DCFFR)	Ours (PCFR+)
B2222	0.00s	0.00s	[1.0]	0.01s	[1.0]	0.00s	0.01s	[1.0]	0.01s	[1.0]	0.00s	0.01s
B2322	0.00s	0.07s	[1.0]	0.22s	[1.0]	0.00s	0.15s	[1.0]	0.34s	[1.0]	0.00s	0.22s
B2323	7.00s	3.75s	[1.0]	7.37s	[1.0]	6.00s	4.69s	[1.0]	9.79s	[1.0]	8.00s	5.39s
B2324	timeout	54.92s	[1.0]	1m 2s	[1.0]	timeout	59.02s	[1.0]	1m 28s	[1.0]	timeout	1m 31s
S2122	0.00s	0.00s	[1.0]	0.01s	[0.1]	0.00s	0.00s	[0.1]	0.00s	[0.1]	0.00s	0.00s
S2123	0.00s	0.01s	[1.0]	0.05s	[1.0]	0.00s	0.02s	[1.0]	0.03s	[0.1]	0.00s	0.01s
S2133	1.00s	0.04s	[1.0]	0.07s	[1.0]	1.00s	0.08s	[0.1]	0.09s	[0.1]	1.00s	0.05s
S2254	1m 41s	8.61s	[0.1]	15.31s	[0.1]	1m 47s	25.38s	[0.1]	23.28s	[0.1]	2m 3s	8.22s
S2264	timeout	20.11s	[1.0]	23.99s	[0.1]	timeout	1m 9s	[0.01]	1m 5s	[0.1]	timeout	16.50s
RS212	0.00s	0.00s	[10.0]	0.00s	[1.0]	0.00s	0.00s	[10.0]	0.00s	[1.0]	0.00s	0.00s
RS222	0.00s	0.00s	[0.01]	0.00s	[10.0]	0.00s	0.00s	[1.0]	0.00s	[1.0]	0.00s	0.00s
RS213	0.00s	0.01s	[10.0]	0.01s	[1.0]	0.00s	0.01s	[1.0]	0.01s	[1.0]	0.00s	0.01s
RS223	0.00s	0.01s	[10.0]	0.01s	[1.0]	0.00s	0.01s	[1.0]	0.01s	[1.0]	0.00s	0.01s
G.3.7 Results for CERT solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize social welfare
LP	Ours (DCFFR)	Ours (PCFR+)	LP	Ours (DCFFR)	Ours (PCFR+)	LP	Ours (DCFFR)	Ours (PCFR+)
B2222	0.00s	0.01s	[1.0]	0.01s	[1.0]	0.00s	0.02s	[1.0]	0.02s	[1.0]	0.00s	0.02s
B2322	2.00s	1.05s	[1.0]	1.16s	[1.0]	2.00s	1.16s	[1.0]	2.29s	[1.0]	2.00s	1.24s
B2323	40.00s	47.11s	[10.0]	1m 2s	[1.0]	33.00s	58.20s	[1.0]	2m 14s	[0.1]	37.00s	46.51s
B2324	timeout	8m 29s	[0.1]	timeout	—	timeout	timeout	—	timeout	—	timeout	6m 14s
S2122	0.00s	0.02s	[0.1]	0.02s	[1.0]	0.00s	0.02s	[0.1]	0.02s	[0.1]	0.00s	0.02s
S2123	1.00s	0.19s	[0.1]	0.37s	[0.1]	1.00s	0.28s	[1.0]	0.31s	[0.1]	1.00s	0.15s
S2133	3.00s	0.96s	[1.0]	1.06s	[1.0]	2.00s	1.10s	[0.1]	1.18s	[0.1]	2.00s	0.92s
S2254	timeout	3m 26s	[0.1]	5m 35s	[0.1]	timeout	6m 23s	[0.1]	6m 15s	[0.1]	timeout	2m 42s
S2264	timeout	timeout	—	timeout	—	timeout	timeout	—	timeout	—	timeout	—
RS212	0.00s	0.00s	[0.1]	0.00s	[1.0]	0.00s	0.00s	[10.0]	0.00s	[1.0]	0.00s	[0.1]
RS222	0.00s	0.00s	[1.0]	0.00s	[10.0]	0.00s	0.00s	[1.0]	0.00s	[1.0]	0.00s	[0.01]
RS213	0.00s	0.02s	[10.0]	0.03s	[1.0]	0.00s	0.02s	[1.0]	0.03s	[1.0]	0.00s	[10.0]
RS223	1.00s	0.01s	[10.0]	0.02s	[1.0]	1.00s	0.02s	[1.0]	0.03s	[1.0]	1.00s	0.01s
G.4 Detailed breakdown by equilibrium and objective function (three-player games)
For each two-player game, we try three different objective functions: maximizing the utility of Player 1, maximizing the utility of Player 2, and maximizing the utility of Player 3. As in Appendix G.3, for each objective we stop the optimization at the approximation level defined as  $1\%$  of the payoff range of the game.
For each game and objective, we report the same information as Appendix G.3.
G.4.1 Results for NFCCE solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize Player 3's utility
LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)
D32	0.00s	0.01s	[1.0]	0.01s	[1.0]	0.00s	0.01s	[1.0]	0.01s	[1.0]	0.00s	0.01s	[1.0]	0.02s	[1.0]
D33	2m 17s	12.93s	[0.1]	43.45s	[0.1]	2m 8s	16.40s	[0.1]	33.57s	[0.1]	2m 23s	17.57s	[0.1]	36.55s	[1.0]
GL3	0.00s	0.01s	[1.0]	0.02s	[1.0]	0.00s	0.01s	[1.0]	0.02s	[1.0]	0.00s	0.02s	[0.1]	0.02s	[0.1]
K35	49.00s	0.76s	[1.0]	0.95s	[0.1]	55.00s	0.56s	[1.0]	1.15s	[0.1]	55.00s	0.64s	[1.0]	1.34s	[0.1]
L3132	26.00s	0.59s	[0.1]	1.09s	[0.1]	24.00s	0.79s	[0.1]	1.28s	[0.1]	24.00s	0.77s	[0.1]	1.18s	[0.1]
L3133	38.00s	0.94s	[0.1]	1.93s	[0.1]	38.00s	0.89s	[0.1]	2.33s	[0.1]	37.00s	1.11s	[0.1]	1.79s	[0.1]
L3151	timeout	15.12s	[0.1]	17.94s	[0.1]	timeout	12.74s	[0.1]	31.74s	[0.1]	timeout	18.03s	[0.1]	31.69s	[0.1]
L3223	4.00s	0.44s	[0.1]	0.92s	[0.1]	4.00s	0.45s	[0.1]	0.98s	[0.1]	4.00s	0.52s	[0.1]	1.26s	[0.1]
L3523	timeout	1m 7s	[0.01]	1m 59s	[0.01]	timeout	1m 2s	[0.01]	1m 44s	[0.01]	timeout	1m 9s	[0.01]	1m 44s	[0.01]
TP3	1m 38s	7.44s	[1.0]	10.71s	[10.0]	1m 40s	7.63s	[1.0]	11.09s	[10.0]	1m 44s	11.45s	[10.0]	11.90s	[10.0]
G.4.2 Results for EFCCE solution concept
Game	Maximize Player 1's utility	Maximize Player 2's utility	Maximize Player 3's utility
LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)	LP	Ours (DCFR)	Ours (PCFR+)
D32	0.00s	0.02s	[1.0]	0.02s	[1.0]	0.00s	0.01s	[1.0]	0.02s	[1.0]	0.00s	0.02s
D33	timeout	1m 46s	[1.0]	4m 31s	[0.1]	timeout	1m 16s	[1.0]	3m 56s	[1.0]	timeout	1m 44s
GL3	1.00s	0.02s	[1.0]	0.04s	[1.0]	1.00s	0.03s	[1.0]	0.04s	[1.0]	1.00s	0.03s
K35	46.00s	0.67s	[10.0]	1.69s	[0.1]	55.00s	0.75s	[1.0]	1.69s	[0.1]	51.00s	0.68s
L3132	8m 43s	5.13s	[0.1]	9.57s	[1.0]	9m 17s	6.27s	[0.1]	12.37s	[0.1]	9m 44s	7.76s
L3133	20m 26s	8.88s	[0.1]	18.30s	[0.1]	timeout	8.19s	[1.0]	18.09s	[0.1]	23m 15s	10.86s
L3151	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
L3223	1m 10s	2.94s	[0.1]	4.85s	[0.1]	1m 10s	3.22s	[0.1]	4.73s	[0.1]	1m 2s	3.24s
L3523	timeout	timeout	[—]	timeout	[—]	timeout	timeout	[—]	timeout	[—]	timeout	[—]
TP3	timeout	13.76s	[10.0]	24.94s	[1.0]	timeout	15.03s	[10.0]	28.28s	[1.0]	timeout	15.27s
G.4.3 Results for EFCE solution concept