Table 7 presents the calculated Q values for simultaneous update. The Q values in Table 7 were calculated in the following way. First we calculated the averages of each column of the payoff matrix in Table 3, see Table 6.
Column | Average |
0.5 | 0 |
0.6 | 0.06875 |
0.7 | 0.1125 |
0.8 | 0.13125 |
0.9 | 0.125 |
1.0 | 0.09375 |
Next, we know that action 0.7 is being played with probability 1-
and other actions are being played with probability
.
We are using an
of 0.05. We used the equation below to calculate each entry in Table 7 using this information. For state x and action y:
state/action | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
0.5 | 0 | 0.0865625 | 0.100625 | 0.0421875 | 0.05375 | 0.0640625 |
0.6 | 0 | 0.0865625 | 0.100625 | 0.0421875 | 0.05375 | 0.0640625 |
0.7 | 0 | 0.0865625 | 0.100625 | 0.0421875 | 0.05375 | 0.0640625 |
0.8 | 0 | 0.0865625 | 0.100625 | 0.0421875 | 0.05375 | 0.0640625 |
0.9 | 0 | 0.0865625 | 0.100625 | 0.0421875 | 0.05375 | 0.0640625 |
1.0 | 0 | 0.0865625 | 0.100625 | 0.0421875 | 0.05375 | 0.0640625 |
state/action | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
0.5 | 0.0001012 | 0.0857952 | 0.101123 | 0.0464701 | 0.056763 | 0.0647148 |
0.6 | 0.000101193 | 0.0859425 | 0.102011 | 0.0465411 | 0.0576224 | 0.0655664 |
0.7 | 0.000101254 | 0.0859014 | 0.100776 | 0.0471251 | 0.0571762 | 0.0655254 |
0.8 | 0.000101306 | 0.0859675 | 0.101669 | 0.0463068 | 0.0564989 | 0.0647921 |
0.9 | 0.000101317 | 0.0859112 | 0.101598 | 0.0453765 | 0.0570052 | 0.0652384 |
1.0 | 0.000101296 | 0.0862614 | 0.101436 | 0.0469018 | 0.0562795 | 0.0658477 |
The pseudo-
decay is,
if | (currRound > 90000000) |
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|
else if | (currRound > 50000000) |
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|
else if | (currRound > 10000000) |
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|
else if | (currRound > 1000000) |
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|
else if | (currRound > 500000) |
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|
else | |
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A. state/action | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|
0.5 | 0.00010024 | 0.0868088 | 0.100121 | 0.0405321 | 0.053697 | 0.0633732 |
0.6 | 0.000100298 | 0.0869526 | 0.100055 | 0.0420465 | 0.0547109 | 0.06311 |
0.7 | 0.00010011 | 0.0876 | 0.1001 | 0.0376307 | 0.0502007 | 0.0626128 |
0.8 | 0.000101017 | 0.0867011 | 0.100106 | 0.042365 | 0.0534665 | 0.0639472 |
0.9 | 0.000101066 | 0.0863434 | 0.100093 | 0.0440427 | 0.0542756 | 0.0637399 |
1.0 | 0.000100977 | 0.0870105 | 0.100091 | 0.0432934 | 0.0522114 | 0.0639043 |
B. state/action | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
0.5 | 0.000100323 | 0.0869103 | 0.100102 | 0.0447314 | 0.0540285 | 0.0636493 |
0.6 | 0.000100411 | 0.0862549 | 0.10007 | 0.0435341 | 0.0545976 | 0.0632749 |
0.7 | 0.000100111 | 0.087599 | 0.1001 | 0.0376873 | 0.0503039 | 0.062612 |
0.8 | 0.000101007 | 0.0865916 | 0.100086 | 0.0401568 | 0.0516071 | 0.0646759 |
0.9 | 0.000101004 | 0.086484 | 0.100055 | 0.0421205 | 0.0563015 | 0.0650803 |
1.0 | 0.000100988 | 0.0867745 | 0.100099 | 0.0409425 | 0.0573791 | 0.0643089 |
Figure 2. Simultaneous, made from Tables 8A and 8B. The arrow indicates the Nash equilibrium of 0.7, 0.7.
When the min price (the other pricebot's price) from the previous round is used as the state for simultaneous, the Q tables converge to all state 7 payoffs as we just saw in Table 9. When the current min price is used as the state, one Q table converges to all state 7 payoffs, the other Q table for the other player converges to the payoff table. See Table 10.
A. state/action | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
---|---|---|---|---|---|---|
0.5 | 0.000106471 | 0.0783584 | 0.106332 | 0.0843278 | 0.0871126 | 0.0776166 |
0.6 | 0.000106473 | 0.0779367 | 0.106332 | 0.0843748 | 0.0878633 | 0.0785019 |
0.7 | 0.000106442 | 0.0779623 | 0.106326 | 0.0842921 | 0.0865126 | 0.0786863 |
0.8 | 0.000106468 | 0.078454 | 0.106564 | 0.0840732 | 0.0880429 | 0.0781503 |
0.9 | 0.000106473 | 0.0783395 | 0.106283 | 0.0844303 | 0.0883922 | 0.0782733 |
1.0 | 0.000106463 | 0.0784538 | 0.106266 | 0.0854391 | 0.0868893 | 0.0780265 |
B. state/action | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |
0.5 | 6.25626e-05 | 0.0125626 | 0.0250626 | 0.0375626 | 0.0500626 | 0.0625626 |
0.6 | 6.25626e-05 | 0.0500626 | 0.0250626 | 0.0375626 | 0.0500626 | 0.0625626 |
0.7 | 0.0001001 0. | 0876001 | 0.1001 | 0.0376001 | 0.0501001 | 0.0626001 |
0.8 | 0.000175175 | 0.0876752 | 0.175175 | 0.150175 | 0.0501752 | 0.0626752 |
0.9 | 0.000262763 | 0.0877628 | 0.175263 | 0.262763 | 0.200263 | 0.0627628 |
1.0 | 0.00035035 | 0.0878503 | 0.17535 | 0.26285 | 0.35035 | 0.25035 |