Sanity Check on Extensive-Form Game in Mathematica Documentation

[Metronome]

Mathematica has functionality which enables extensive-form games to be constructed. It is possible to input probabilities onto edges, as shown in the second image in the Details and Options section on this page, and pasted below...

• Additional information for action nodes include action labels and action probabilities:

• 	{playerj,…,actioni->labeli,…}	using labeli for actioni
  	{playerj,…,actioni->probi,…}	choosing actioni with probability probi
  	{playerj,…,actioni->{labeli,probi},…}	specify both label and probability

• When using probabilistic actions at an action node, then all actions need to have probabilities for that action node.

I can't quite figure out the semantics of this. It doesn't seem to be that of a chance node, since player1 (and not nature) is assigned the node prior to the edges with probabilities. It also shouldn't be strategic mixing, since game trees are typically constructed and stored (logically) prior to, and decoupled from, players' strategies. Indeed, there is other functionality which asks for a mixed strategy and a game tree as separate inputs (and checks whether the strategy is a subgame perfect equilibrium). And the edges with probabilities are not directed into the payoffs, so do not appear to be uncertainty about a player's type as in a Bayesian Game.

Am I correct in saying that this idea of a real player such as player1 making a choice into game tree edges with probabilities is absurd, or is there some other concept from game theory I'm not aware of being instantiated here?

 

[Cubist]

Looking at the documentation, it seems that supplying probabilities on edges is optional:-

> When using probabilistic actions at an action node, then all actions need to have probabilities for that action node.

Of course, if every edge has been assigned a probability then it would be a game without choice. For example snakes and ladders. Actually there is one choice in this game (at the very start) should I play the game or not!

 

[Metronome]

Looking at the documentation, it seems that supplying probabilities on edges is optional:-

> When using probabilistic actions at an action node, then all actions need to have probabilities for that action node.

Of course, if every edge has been assigned a probability then it would be a game without choice. For example snakes and ladders. Actually there is one choice in this game (at the very start) should I play the game or not!

Sure, it is indeed optional. Maybe this is just a different convention than I'm used to. In a game tree for Snakes and Ladders, I wouldn't have thought it correct to write "player1" or "player2" on any of the nodes (rather, all moves would be made by a fictional player "Nature," or "Chance," or "God," etc.), but I could also see why this discards useful information for such a game. In general, of course, a game need not have such a clean isomorphism between nodes and turns, or between nodes and changes in the state of a character or game piece "owned" by a particular player.

 

[Cubist]

The label seems arbitrary. I don't think that it matters to this "TreeGame" function...

Personally, I would definitely label each node with something much more descriptive. Something that fully describes the state. Perhaps I would go for "Player 2 to move, player 1 on square X, player 2 on square Y". I think that this would completely describe the node.

BUT, it looks like there's no way to form any type of graph using this function. We can't link back to nodes "above" the current move "row". I guess that's why it's called "TreeGame". We must always propagate down to new states (we can't go back to a previous row). Therefore this could never represent the game of "snakes and ladders" where a previous state could be arrived at again (by sliding down a snake).

Sorry, I am learning too... I have never used this function. I'm just interested in game theory!