Understanding the properties of extensive form games

In Heller et al, they use the Osborne and Rubinstein formal definition for the extensive form games with public information. To some point they refer to the following two properties

• $$P$$ is a mapping that assigns to each non-terminal history $$h$$ the set of players $$P(h)\subseteq I$$ that have to take an action after history $$h$$. If $$P(h)= \emptyset$$, then there is a chance move after history $$h$$.

$$\textbf{Question 1:}$$ What does this prhase mean: $$P(h)= \emptyset$$, then there is a chance move after history $$h$$"? Is it like the game is played upon some graph?

• $$A$$ is a mapping that assigns to every non-terminal history $$h$$ such that $$P(h)\neq ∅$$, and to every player $$i \in P(h)$$, a finite set $$A_i (h)$$ of actions available to player $$i$$ after that history. Let $$A(h)$$ be the set of available action-profiles at $$h$$: $$A(h) = ×i∈P(h)$$ $$A_i (h)$$. If $$P(h)=∅$$ for some non-terminal history $$h$$, then $$A(h)$$ is the finite set of chance moves at the history $$h$$.

• $$f$$ is a mapping that assigns to every non-terminal history $$h$$ such that $$P(h)=\emptyset$$, a probability distribution $$f(·|h)$$ over chance moves $$A(h)$$. That is, when chance has to move after a non-terminal history $$h$$, an action $$a \in A(h)$$ is chosen according to the probability distribution $$f(·| h)$$.

$$\textbf{Question 2:}$$ Could anybody give an example about how these three properties work for some game?

• Prisoner's dilemma and Chicken game are simultaneous move games, so their extensive form would be rather atypical. Also, they don't have chance moves. Nov 15 '21 at 10:36
• @VARulle then think of some game.... I will change the question. Nov 15 '21 at 10:45
• It's not yet clear what the question really is. These points are not "properties" of a game but the definition of an extensive form game with public information. Nov 15 '21 at 12:43
• @VARulle ok1 Then the question is, show us with an example what each one of these means? The defitiotion has these three parts. Tell as what you understand from them? Can you give an example so as we could understand how the game works too? Nov 15 '21 at 14:00

So $$h$$ is just some history of the game. Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities depends on the choice of Player 1.

An example of a history $$h$$ is $$h = Heads$$, i.e. after heads is chosen.

Consider the very beginning of the game, before Player 1 has chosen anything, it is the empty history $$h = \emptyset$$.

By the definition above, $$P(\emptyset) = 1$$, since player 1 moves there, $$P$$ tells us who moves there.

On the other hand, consider $$h = Tails$$, so the node right after Player 1 has chosen Tails.

(i) Only nature moves (NOT a player!) so then $$P(Tails) = \emptyset$$.

(ii) Now, $$A(h)$$ tells us what moves are available to whoever is moving at that point. So for our example, $$A(\emptyset) = \{Heads,Tails\}$$, and $$A(Heads) = \{1,2\}$$.

(iii) Finally, $$f$$ just tells us what the probabilities of nature's moves. So $$f(1 | Heads) = Pr(\text{1 happens | Heads}) = 3/4$$.

For completion, $$f(3 | Tails) = 1/3$$.

To summarise, $$P$$ tells us who moves at a node. If $$P(h) = \emptyset$$, Nature is moving.

$$A_i(h)$$ tells us the moves available to Player $$i$$ at node $$h$$. In the case nature is moving, it is just the available outcomes of nature's randomisation, and the randomisation occurs via the probabilities described by $$f(\cdot | h)$$.

• You mean that when $h=\emptyset$, in other words at the starting point, then the players who moves is player $1$ namely $P(\emptyset)=1$? Nov 15 '21 at 17:31
• Yessir, indeed. Nov 15 '21 at 18:06
• thank you. In essence, this player function tells us, after histroy $h$ has moved forward, which players still remain in the game. Maybe, there are players that do not go on playing the game infinitely for many reasons. However, this chance move function is an inovvation that I have not seen anywhere else. I can not understand what is the intuition behind this... Nov 15 '21 at 19:15
• I would say $P$ just tells you who moves at that point in time, it does not tell you about future events. I'm surprised you have never seen chance moves on a game tree, its quite standard. Nov 15 '21 at 21:36
• You either stay home and get 0 payoff or you go for a walk. If you go, it will be rain or sunshine with prob. 1/2 each and payoffs -1 and 1, respectively. That's a "move of nature" or a chance event, and by definition $P(go)=\emptyset$. I think that's very natural and the "intuition behind this" is not mysterious. Nov 16 '21 at 10:34