The title-question is the main focus of my confusion but I have several supporting questions at the end that try to clarify.
To illustrate my question, I will base it off of a game I played a while back during a team-building exercise at a university camp. I can't remember exactly how the rules were phrased. Basically all the students were blindfolded and led to a place where we were given a singular, long-taught rope to hold.
These were the constraints that were given to us or that could be inferred:
- We were told that when the whistle blew we could begin moving.
- We could move anywhere in space as long as we were touching the rope at any given moment (it was possible to pass other people with some talking and physical co-ordination).
- If we thought we'd found the end of the rope we could raise our hand and some authority would tell us whether we had actually found it. There was no limit to how often we could do this.
- If we found the end of the rope, we won.
- We could ask to surrender; the game ended for us and we left with our blindfolds off.
It might not be the best formulation, but I have decided to model it as a one-player game with two actions, namely: surrender or continue searching for the end of the rope. Let $k$ be a quotient representing how much patience and/or resolve we have. And let $t$ be how much time has elapsed in minutes. Given the following decision matrix.
$$\begin{array}{c|c|}
\hline
Surrender & 0\\
Continue & 15-(t/k)\\
\hline
\end{array}$$
As time went on in the game, I heard as people asked if they had found the rope and being told that they hadn't. Soon, people with lower patience $k$ started surrendering. Everything fit the model of the rules explained so far.
The problem arose when I reached a knot in the rope. I recognised the knot from my starting point and realised that I had come full-circuit. The implication of this was that the theoretical win-condition stated at the beginning was un-achievable.
Much later, after the game, the facilitator told us that the solution was to surrender. The game was supposedly designed to show us how self-dependent we try to be. Before that, however, after I had realised that the actual rules were different to the way I had perceived them, I tried to formulate a new game with my new information. Since I wasn't satisfied to end with a solution that gave me the minimum utility.
Therefore, considering I was effectively playing against the patience of the facilitators, I adapted it as minimally as I could to a two-player game like the utility matrix bellow (where $j$ is the patience and/or resolve of the facilitator).
$$\begin{array}{c|c|c|}
& Surrender & Continue\\
\hline
Surrender & 0,0 & 0,5\\
Continue & 5,0 & 15-(t/k),15-(t/j)\\
\hline
\end{array}$$
The logic was that if I could motivate myself to have $k > j$, then I would have to win. Which, if this game was the measure, I eventually did, to the annoyance of the facilitator.
- So what I want to know is, what was the actual game?
- At what point did the rules change (if at all)?
- If they did change, was my adaptation from the original one more reasonable or did the facilitator's "admission" about lying about the win condition not remove credence from their new win condition?
- Is there a better way to model the two games (together maybe)?
- This all borders on philosophy because it is unclear who determines the rules but I basically want to know what the "mathiest" way to talk about it is.
If there is any way this question can be clearer, feel free to edit it or suggest a change. I am just a hobbyist mathematician.
Best Answer
When you're modeling a game, it's all about identifying strategies and outcomes. I think in your case, both of these are a bit underbaked; here's what I mean by that.
Strategies
When identifying strategies available to each player, you must be exhaustive. Every action that's available to each player at any point in the game must be represented somehow. In the simple game you've written above, there are two strategies: Surrender and Continue. But these don't represent the sum total of actions available to you, for instance you could continue until finding the knot and then surrender. Or you could surrender immediately. Or you could continue forever. Or you could spend up to 10 minutes searching, and then surrender. Likewise, the facilitator has many options.
I see two natural ways to capture this without changing your framework too much: continuous strategies and repeated play. For the first, one approach might be to define two new variables, $d_s$ and $d_f$, where $d_s$ is the total duration of time you're willing to search, and $d_f$ is the duration the facilitator is willing to let you search.
For repeated play, you could model the game as a series of decision points, at each one the players have the option to stop the game or let it continue. If the actions available at each decision point are the same, then the game can be represented by the single matrix you have, and some analysis can be done on the options available to the players. In the end, a "meta"strategy in the repeated game is going to look a lot like the continuous strategy above, since if at any point either player surrenders the game is over.
Outcomes
I think another source of difficulty is that you've tried to jump straight to the payoffs for each player. It's usually sounder to start by explicitly listing the outcomes, and then assigning payoffs to the outcomes afterward. For instance, some of the possible outcomes here are: The game ends quickly and both players are satisfied; the game drags on and both players are dissatisfied; the game drags on and the searcher feels vindicated but the facilitator is annoyed, etc., etc., and the actual length of the game may have some bearing on how satisfied / dissatisfied the players are.
The considerations about whether the rules changed, what it means that the facilitator lied about the nature of the game, etc., can (and should) be completely accounted for in the outcomes, and thus in the payoffs. For instance, in the continuous strategy example I gave above, in the case that $d_s > d_f$, I get the sense you would call that a victory for the searcher, so the payoffs may be $(10-d_f, -d_f)$ to indicate how annoyed you are that the facilitator made you play for so long, and how annoyed the facilitator is that they didn't get their preferred outcome and it took so long.
One thing that's very important to keep in mind is that in this kind of game theory, the concept of "winning" is basically meaningless. There is only the value (utility) that each player gets from whatever outcome occurs. This is one of the most difficult mental hurdles new game theorists must get over.
I hope this helps you think about the problem!