Thanks for answer. You are in my blacklist due to spamming in chat! Nothing more.

1 common game:
https://www.warzone.com/MultiPlayer?GameID=13630779

Whatever.

@Norman
Regarding the first example, you should consider your conclusions in a wider frame. There is not a specific point about one being the attacker and one the defender.

The point is that there is a player that has an advantage (in your example the defender, as in order to win, he needs to do the correct move once in two turns) and the general rule is that:
“The player that has an advantage plays more often the strategy that grants the best worst expected result” (more simply plays more often the safer or obvious move)

Example: there is a choke where Player1 has stack advantage and needs to choose between:
A) Fulldeploy where he already has stack advantage and break his opponent bonus 100% granted, while compromising his secondary objective (expansion, another border, whatever…).
B) Try a smaller attack (and risk being defended), while pursuing also his secondary objective.

Assuming that Player 1 winning chances are as follows:
- Option A: 70% if Player2 defends and 60% if Player2 does not
- Option B: 50% if Player2 defends and 90% if Player2 does not

then Player 1, being in advantage should play more often Option A that “grants the best worst result” (i.e. 60% win rate vs 50% of Option B).

This is true despite Option A seems worse as average, and would apply even with extreme % as:
- Option A gives 51% or 52% win rate depending on Player 2 move
- Option B gives 50% or 100%.
The strategy that cannot be outplayed is still picking Option A (actually surprisingly close to 100% of the times).
---

As of the picking question, under perfect information assumptions, this is a plain rock/paper/scissor game so the correct way to play is 1/3 for each choice: (1) dominant pick, (2) counter and (3) other picks (assuming this loses 100% vs dominant, wins 100% vs counter and gives 50% vs any combination of other picks).

Edited 11/11/2019 20:42:31

Aside from the theory above, coming to the point of "Optimal Play", this is by far more complicated.

The very first consideration is that playing the theoretical % calculated with Nash equilibrium will give nothing more than average results against weak or strong players.

E.g. a bot programmed this way may score below a strong human in a round robin tournement, as being unable to take benefit outpredicting weak players.

So a possible rule is: play the theoretical chances vs players that are stronger at predicts and try outpredict weaker players.

@Hergul: For your scenarios I get the following results:

Your first input:
-----
The matrix is
0.9 0.5
0.6 0.7
The value is 0.66.
An optimal strategy for Player I is:
(0.2,0.8)
An optimal strategy for Player II is:
(0.4,0.6)
-----

Your second input:
---
The matrix is
1 0.5
0.51 0.52
The value is 0.51961.
An optimal strategy for Player I is:
(0.01961,0.98039)
An optimal strategy for Player II is:
(0.03922,0.96078)
---

(The rows are Player 1s decision and the Columns Player 2 decision. The first entry means no deployment and the second means deployment)

The results here are according to what you wrote, for example the second results mean that Player 1 has to deploy 98% of the time and Player 2 has to defend 96% of the time.



The very first consideration is that playing the theoretical % calculated with Nash equilibrium will give nothing more than average results against weak or strong players.

E.g. a bot programmed this way may score below a strong human in a round robin tournement, as being unable to take benefit outpredicting weak players.

So a possible rule is: play the theoretical chances vs players that are stronger at predicts and try outpredict weaker players.

Exploiting weaknesses can obviously increase your win chances. However I'm not completely sold on your first point that playing optimally without exploiting the opponent only gives average win chances. I'm thinking for example at 3v3 Europe which is liked by strong players due to it not being very "rock paper scissors" like. However the optimal picking strategy is very difficult to see on this template. I got outpicked quite some times here where we had like the first 12 picks completely identical and with the later picks we had only slight differences in the order. In this scenario my picks were just plain worse than my opponents, so an optimal player would never go for my picks. However a team playing optimal will maybe never prioritize Spain high but might from time to time go for a sneaky Poland.

Also there are players who were capable of winning the Seasonal Ladder up to 3 times. This confirms my assumption that in an average WarLight turn you have a lot of moves which seam feasible to an average player however the higher the skill level goes the more the players can narrow the moves down since some moves are strictly dominant over the others. For example a highly skilled player might understand after picks that if his opponent counters him in a certain way he has already lost, so for this reason the optimal play is to play as if the possibility of this opponent counter is 0.

Thanks for giving the precise math of my examples. What I also mean is that your first example where you mention the roles of defender and attacker falls under the same wider logic of "party in advantage", that in your example is the defender, having two options:
1) Defend the bonus: 100% win, 50% win
2) Defend the side: 100% win, 0% win
And the "optimal play" is to go for opt. 1 more often, according to the general rule I referred to.

Regarding the other point, i.e. that "Optimal Play" gives average chances, I expressed the concept poorly. What I mean is that in the rigid frame where there is a set of options none of which is dominant or losing, a bot programmed for what we call "Optimal Play" will win an average number of times according to the specific situation. E.g. "Optimal Play" in a 50/50 situation will win 50% of the times, even vs a weak player that always picks an option and is easily outplayed by any decent player.

Hence the second rule stands, i.e. "use Optimal Play vs players stronger than you (e.g. flip the coin for 50/50 decisions), try to outpredict weaker players".

Your example about the 3vs3 Europe map does not fall in the rigid boundaries of the theorical examples, where I assumed perfect information and no dominant/losing option.

I actually fully agree that stronger players outplay others because of two reasons:
1) Are stronger in evaluating possible alternatives (moves, picks, whatever) and the asimmetry in the information
2) Are stronger at predicting

And in my view point 1 is by far the most important. I have analyzed many games I lost vs strong opps, and very often I learned they just used superior strategies and not predicts.

"I actually fully agree that stronger players outplay others because of two reasons:
1) Are stronger in evaluating possible alternatives (moves, picks, whatever) and the asimmetry in the information
2) Are stronger at predicting

And in my view point 1 is by far the most important. I have analyzed many games I lost vs strong opps, and very often I learned they just used superior strategies and not predicts."

Strongly agree with this. I've played a great many games, some at very high skill level. Some involved chance, some involved predictions, all involved strategy. In all those games winning with strategy is preferable to relying on chance or predictions. The latter can win games, but good players put themselves in positions where those don't matter, because it is far more consistent.

Edited 11/22/2019 18:11:12

My (1 v 1 experience mostly on the official ladder maps) 2 cents to be added in this wonderful conversation...that I followed for 3-4 posts so excuse me if the following have been already discussed....

Warlight resources (numbers) can be spent on two categories. The first one is FIGHT and the second is EXPAND. Given similar skills of opponents that include the ability to properly analyse the board and conclude which of the two above aspects they should spent their numbers on, the one who manages to spend LESS ON FIGHT => MORE ON EXPAND than the other, should win.

I prefer to keep it as simple as possible so I'll leave it there, knowing that resource allocation is a very long story.