Weerwolven van Wakkerdam
Implementation and analysis
We modeled the game using Java, and implemented the strategies discussed in the previous section. For a detailed view, see the code and the comments therein. We ran games with different numbers of players with different strategies, and analysed whether our model corresponded to the theory. In the next sections, we consider the different situations.
One seer, one werewolf, different number of citizens, all zeroth-order
The results of the runs are shown in the following table.
| citizens | wins of werewolves | wins of citizens |
|---|---|---|
| 1 | 502 | 498 |
| 2 | 667 | 333 |
| 3 | 384 | 616 |
| 4 | 539 | 461 |
| 5 | 320 | 680 |
| 6 | 457 | 543 |
| 7 | 284 | 716 |
| 8 | 419 | 581 |
| 9 | 234 | 766 |
| 10 | 363 | 637 |
| 29 | 140 | 860 |
| 30 | 220 | 780 |
With one zeroth-order citizen, one zeroth-order seer and one zeroth-order werewolf, the results are completely random. First, the werewolf eats one of the other two players. Then the resulting two players vote for each other, and becasue there is a random person selected if the voting result is equal, both factions have the same chance to win.
If there are two citizens, one seer and one werewolf, there are two of the citizens and seer left in the day-phase, and one werewolf. The chance that the werewolf will be burned is generally lower than 0.5: the werewolf doesn't vote for himself, so if the seer knows the werewolf the werewolf and the seer vote for each other, the result is dependent on the ignorant citizen, and the chance will be 0.5. However, if the seer doesn't know this or is already dead, the chance that a citizen is burned is larger, and the werewolf can easily eat the last citizen in the night-phase.
With three citizens, the specific circumstances described above are not applicable, the werewolf will win much less times.
From the table, it appears that with an even number of citizens, the chance to win is almost the same for both factions. This is probably due to the circumstances described in the two-citizens case. For an odd number, the chance to win for the werewolves decreases if the number of citizens increases. For large numbers of citizens, however, the chance for the werewolves to win also decreases with an even number of citizens.
One seer, two werewolves, different number of citizens, all zeroth-order
The results of the runs are shown in the following table:
| citizens | wins of werewolves | wins of citizens |
|---|---|---|
| 1 | 1000 | 0 |
| 2 | 888 | 112 |
| 3 | 931 | 69 |
| 4 | 775 | 225 |
| 5 | 867 | 133 |
| 6 | 664 | 336 |
| 29 | 471 | 529 |
| 30 | 361 | 639 |
In the situation with one citizen, the werewolves will eat the citizen or the seer and burn the other. So they will always win. In this case, this is a good strategy.
When the number of citizens increases, they will win more often, following the pattern described in the previous section.
The influence of the seer, all zeroth-order
To test what the influence of the seer is, we conducted the following tests:
| citizens | seers | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|
| 1 | 1 | 1 | 502 | 498 |
| 2 | 0 | 1 | 506 | 494 |
| 7 | 1 | 1 | 285 | 715 |
| 8 | 0 | 1 | 257 | 743 |
| 7 | 1 | 2 | 781 | 219 |
| 8 | 0 | 2 | 782 | 218 |
From these results, we conclude that the influence of the seer is relatively small in the case of only zeroth-order players. This is because citizens do not check if other players correctly vote for werewolves (like seers would more often do), so citizens still vote for seers to be burned.
One faction first-order, the other zeroth-order
To test whether one group has an advantage when thinking in a higher order, we conducted the following experiments:
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 5, 0th-order | 1, 0th-order | 1, 0th-order | 2, 1st-order | 578 | 422 |
| 5, 1st-order | 1, 1st-order | 1, 1st-order | 2, 0th-order | 365 | 635 |
| 5, 0th-order | 1, 0th-order | 0, 0th-order | 1, 1st-order | 272 | 728 |
| 5, 1st-order | 1, 1st-order | 0, 1st-order | 1, 0th-order | 315 | 685 |
In the first two rows above, the higher-order groups seem to gain an advantage from their higher-order thinking. However, when there is no innocent girl the werewolves do not profit from higher order reasoning because they try to identify an innocent girl when there isn't one. We conclude that it's highly dependent on the composition of the group.
With larger groups, however, the advantage of the higher-order thinking group is more manifest; see the following table. Players with higher order reasoning have many oppertunities to build up their knowledge base and all roles are present. This means that the werewolves can easily identify innocent girls and eliminate them, since only innocent girls will consistently vote for werewolves.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 15, 0th-order | 1, 0th-order | 1, 0th-order | 3, 1st-order | 715 | 285 |
| 15, 1st-order | 1, 1st-order | 1, 1st-order | 3, 0th-order | 11 | 989 |
| 15, 0th-order | 1, 0th-order | 1, 0th-order | 5, 1st-order | 985 | 15 |
| 15, 1st-order | 1, 1st-order | 1, 1st-order | 5, 0th-order | 41 | 959 |
Comparison all first-order vs. all zeroth-order
When we compare first-order with zeroth-order players, we get the following results. It seems that if all players use the same order, the first-order behavior is better for the citizens than the zeroth-order.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 5, 0th-order | 1, 0th-order | 0 | 1, 0th-order | 320 | 680 |
| 5, 1st-order | 1, 1st-order | 0 | 1, 1st-order | 303 | 697 |
| 15, 0th-order | 1, 0th-order | 1, 0th-order | 3, 0th-order | 642 | 358 |
| 15, 1st-order | 1, 1st-order | 1, 1st-order | 3, 1st-order | 133 | 867 |
This can be explained by the fact that the knowledge updates of the citizens in the first-order with respect to the zeroth-order are much more important than the new knowledge update of the werewolves.
One faction second-order, the other first-order
We compare a second-order group with a first-order group.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 10, 1st-order | 1, 1st-order | 1, 1st-order | 2, 2nd-order | 483 | 517 |
| 10, 2nd-order | 1, 2nd-order | 1, 2nd-order | 2, 1st-order | 197 | 803 |
Here, we see that the higher-order knowledge group has a great benefit. This is the same in the following experiment.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 0 | 5, 1st-order | 0 | 3, 2nd-order | 758 | 242 |
| 0 | 5, 2nd-order | 0 | 3, 1st-order | 695 | 305 |
One faction second-order, the other zeroth-order
In the following experiment, we investigate the influence of one faction being two orders higher than the other. In theory, the results will be less good than with a one-order difference. For the werewolves, this is the case, compared to the experiment in the previous section. For the citizens, however, the two-order difference gives better results than the one-order-difference.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 10, 0th-order | 1, 0th-order | 1, 0th-order | 2, 2nd-order | 265 | 735 |
| 10, 2nd-order | 1, 2nd-order | 1, 2nd-order | 2, 0th-order | 69 | 931 |
One faction third-order, the other second-order
We compare again two factions with a one-order difference.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 10, 2nd-order | 1, 2nd-order | 1, 2nd-order | 2, 3rd-order | 294 | 706 |
| 10, 3rd-order | 1, 3rd-order | 1, 3rd-order | 2, 2nd-order | 423 | 577 |
In contrast to the first vs. second order, wee see in this case that the lower-order knowledge group has a benefit. Apparently, there are too much influences to let the strategies work.
One faction third-order, the other zeroth-order
Here we see that third-order vs. zeroth-order will not have a benefit for the third-order group.
| citizens | seers | innocent girls | werewolves | wins of werewolves | wins of citizens |
|---|---|---|---|---|---|
| 10, 0th-order | 1, 0th-order | 1, 0th-order | 2, 3rd-order | 204 | 796 |
| 10, 3rd-order | 1, 3rd-order | 1, 3rd-order | 2, 0th-order | 279 | 721 |
Conclusion
In general, the following aspects are the most important and notable.
Werewolves have mostly knowledge, citizens have mostly beliefs, but in our implementation, the difference does not matter.
With zeroth-order knowledge, the players have just initial knowledge about the world; in the first-order, they are reasoning about what roles other players have. Second-order players try to mislead opposing players about what role they have themselves and third-order players try not to be mislead by what role other players have. However, they are misleading themselves. Fourth-order players try not to be mislead by what role other players have and are no longer misleading. The fifth order is comparable to the first, the sixth to the second, etc.
Citizens rely heavily on reasoning and heuristics to optain knowledge about who is werewolf, but werewolves always already know who is citizen and who is werewolf. Therefore, citizens benefit more from higher order reasoning than werewolves do. This was also shown by the experiments. Increasing to higher order reasoning therefore equals the playing field a little bit between citizens and werewolves.
It is possible to be too smart: if you play as if your opponent has advanced strategies, yet your opponent is actually dumb, then you may make the wrong moves. For example, our results show that second-order werewolves get better result with respect to first-order citizens than with respect to zeroth-order citizens.
As knowledge orders increase, the effect of each increment becomes smaller and smaller. The step from doing nothing (zero-order) to doing at least something (first-order) is the most important one.
When players try to deceive each other by sometimes acting randomly, it is difficult for higher orders to respond effectively because it is hard to draw reliable conclusions from behavior that contains random elements. Therefore, we can conclude that deceptive behavior is generally very effective against opponents that are at least somewhat smart.
So, our model follows in general our theory, but due to random behavior and the complexity of all strategies, the results become less predictable in more complex situations.
Future work
In this game, it's probably interesting to use some kind of probabilistic logic to model to which extent players belief things. To some extent, this is already implemented: for instance citizens consider players that previously voted on citizens to be wolves with a higher probability than the average player. However, using a more thorough probabilistic model players could also take voting behavior of other players into account for their own voting behavior: if they consider a certain player to be a like seer, then they might want to emulate the voting behavior of that seer. However, because nothing is certain, it would require a lot of probabilistic calculations to determine the optimal course of action. If the seer is suggesting something, it could be considered more reliable than an accusation from another citizen. Not only the distinction between knowledge and beliefs is interesting, but also the differences between different kinds of beliefs. Some rules would lead to stronger beliefs.
Another interesting point is the discussion before the burning-ceremony in the day-phase. In the game, the players can debate at that time to convince each other about their beliefs and knowledge. In reality, voting behavior is strongly influenced by this debate. An interesting point to model is this debate, wherein people can base their opinion about the statements on their beliefs about the characters of the players who are making these statements. However, we expect the social model to be very complex. At the very least, it would require probabilistic logic.
In a more advanced model, the players probably can adjust their order on the basis of the behavior of the other players. However, if for example different citizens have different order, finding an appropriate strategy becomes more and more complex for a werewolf.