# Follow Up On The Queuing Order Article

by MannySkull

Yesterday I wrote an article that intended to introduce concepts from game theory that are relevant to understand static games. I used these concepts to describe how to choose a deck (out of three) in the first round of a Bo5 setting. In my mind, the plan was to build up from the first queuing decision (which is the simplest one if you do not take into account future turns) and then build up to the Bo5 case, explaining how each additional layer was going to affect (or not) the decision that was optimal in isolation. That was the plan and, to be clear, I was going to execute the plan as I was writing the articles (that is, I had no idea what the end solution was going to be). That was the plan.
Then, several readers criticized the article. First, some complained that the article was not a comprehensive treatment. Well, sure. Second, some complained that it was so basic that it was useless to understand the entire Bo5. Well, to me it is always best to start simple and build up.
And a bunch of other criticisms, many of them valid and relevant and so others less so. However, there was one that called my attention: the claim that if your opponent is randomizing decks with equal probability (that is, using the random number generator from random.org) then that strategy cannot be exploited by you, the player. This was interesting because it does not happens in a static game where the mixing probabilities depend on the payoff values (matchup probabilities). It also meant two important things: (1) my idea of building up from Bo1 to Bo5 was flawed and was not going to lead anywhere; (2) conquest Bo5 (and Bo3) is a type of game in which the payoff matrix (matchup probabilities) affect your odds of winning but those odds are identical for any possible queuing order (again, when your opponent uses random.org). Finally, if that’s the case, then both players using random.org is an equilibrium of the game. So, last night I grabbed pen and paper and worked out the algebra for Bo3 and Bo5 and concluded that:

1. I was wrong in the plan developed by the first article. Understanding the equilibrium when the problem is simply two players queuing a deck is irrelevant for understanding the Bo5 solution. To be clear, if you want to understand how to think about a 3×3 static game, see what a dominated strategy is, etc, the previous article is not wrong and you may find it useful: it simply does not serve the purpose I original intended for the article.
2. In Bo3 and Bo5, if your opponent uses random.org (i.e., the opponent randomizes/mixes decks with equal probability), then the ex-ante odds of you winning the round are independent of the order in which you queue your deck so you cannot exploit your opponent’s behavior. It follows from here that if you also use random.org, we have an equilibrium where both players are happy.