<p>Your question is a little bit vague, but I'll do my best. </p>

<p>Your professor was motioning in the correct mathematical direction, but I think that what you're asking for is a more complete formalization.</p>

<p>Any game can be thought of as a tree (i.e. a non-cyclic graph). So to answer your first point about turn-taking, you might think about a function which maps nodes of the tree to some $i\in\{1,2,...n\}$ In the picture below, you might be able to define a formula to characterize the turn taking, but you can't necessarily do this in general. In this example $P_1$ and $P_2$ take alternating sequential turns. $P_1$ can choose $L$ or $R$, then $P_2$ can see which he chose, and then also choose $L$ or $R$.</p>

<p><img src="https://i.sstatic.net/4cNLP.gif" alt="Game Tree Example"></p>

<p>Before going on to your second point, I need to take a moment to define information sets, which it doesn't sound like you've gotten to yet. Colloquially, an infoset is a set of vertices that represent uncertainty. The root of a game tree is always a singleton infoset, for example. Consider the game where you and I simultaneously choose heads or tails. When $P_2$ chooses whether or not to play heads or tails, she doesn't know if I have chosen to play heads or tails either. Thus while the game tree will have 3 internal vertices (1 that is the root for $P_1$, and 2 for $P_2$, reflecting $P_1$'s choice) there will only be 2 infosets because both vertices for $P_2$ will be contained in the same infoset. In games where turn-taking is sequential, all infosets should be singleton. </p>

<p>Now back to your question about strategies. Each player has a set of strategies. In the example tree provided above $S_1 = \{LLLLL,LLLLR,...,RRRRR\}$. This might seem a little bit confusing at first as it seems like each element of $S_1$ goes above and beyond defining a path. Each player's strategy is defined in terms of infosets. Therefore, the element $LLLLL$ should be read as, when $P_1$ finds herself at the top, she will choose $L$, and when she finds herself at the vertex that is farthest on the left and 1 down from the top, she will also choose $L$. It then also says that if $P_2$ were to choose $R$ instead, that $P_1$ would choose $L$. Thus a strategy is the set of all possible actions a player might take under all possible contingencies.</p>

<p>Finally, how do we state the mapping from strategies to payoffs? First we note that ever player has a strategy set $S_i$, and then form the cartesian product $S=S_1 \times S_2 \times ... \times S_n$, which characterizes all possible strategy combinations for all possible players. Then our payoff will be $\pi:S\to\mathbb{R}^n$. In our example $\pi(LLLLL,LLLLLLLLLL) = (4,2)$. </p>

CC BY-SA 3.0
2015-02-14T18:23:35.623Z
2015-02-14T18:23:35.623Z
Galen Harrison
<p>My professor defined a <em>game</em> as follows: </p>

<ul>
<li>a set of $n$ players </li>
<li>a set of strategies $s_i$ for each player $i$</li>
<li>a set of payoffs $p_i$ for each player $i$</li>
<li>turn-taking: sequential or simultaneous </li>
<li>repetition: $k$ rounds, where $k\geq1$ and $k \in \Bbb{N}$</li>
</ul>

<p>I have three problems with this definition:</p>

<ol>
<li>He didn't define turn-taking mathematically, so I feel like it's a vague definition. There should be like an index of possible turns with max value $n$. But I'm not sure how to write it formally. </li>
<li>He doesn't define how we map from strategies selected to payoffs. Presumably, the set of payoffs should include a value for every possible combination of strategies selected. But there should be a way to state this formally as well.  </li>
<li>He doesn't define what a strategy is. I suppose one could simply say that a strategy is the object we map from in the domain to an element in the set of payoffs (ie codomain). In that sense, anything that can be valued <em>is</em> a strategy, ie a possible action. </li>
</ol>

<p><strong>My Question:</strong></p>

<p>Can someone offer a more complete definition than the one given? Specifically make sure it improves upon the concerns I mentioned. </p>

CC BY-SA 3.0
2015-02-17T17:12:42.993Z
You can try one of many textbooks, there is one by Fudenberg and Tirole (not my personal choice), Myerson, and many others. You can find Osborne and Rubinstein online here: http://books.osborne.economics.utoronto.ca/
usul
CC BY-SA 3.0
2015-02-17T17:27:08.900Z
Great, that's even better. I didn't know where to look. Thanks.
Stan Shunpike
CC BY-SA 3.0
2015-02-18T20:11:36.597Z
The definition of a game varies depending on the *kind* of game in question, e.g. extensive vs normal form, complete vs incomplete information, etc. Besides Osborne-Rubinstein, I'd also recommend Mas-Colell, Whinston, & Green (aka MWE).
Herr K.
<game-theory>
Help improving this definition of a "game"?
Stan Shunpike
# Help improving this definition of a "game"?

(Asked by: username_0 on 2015-02-14)

My professor defined a *game* as follows:

* a set of $n$ players
* a set of strategies $s\_i$ for each player $i$
* a set of payoffs $p\_i$ for each player $i$
* turn-taking: sequential or simultaneous
* repetition: $k$ rounds, where $k\geq1$ and $k \in \Bbb{N}$

I have three problems with this definition:

1. He didn't define turn-taking mathematically, so I feel like it's a vague definition. There should be like an index of possible turns with max value $n$. But I'm not sure how to write it formally.
2. He doesn't define how we map from strategies selected to payoffs. Presumably, the set of payoffs should include a value for every possible combination of strategies selected. But there should be a way to state this formally as well.
3. He doesn't define what a strategy is. I suppose one could simply say that a strategy is the object we map from in the domain to an element in the set of payoffs (ie codomain). In that sense, anything that can be valued *is* a strategy, ie a possible action.

**My Question:**

Can someone offer a more complete definition than the one given? Specifically make sure it improves upon the concerns I mentioned.

**Tags:** game-theory

> **Comments:**
> **username_1**: You can try one of many textbooks, there is one by Fudenberg and Tirole (not my personal choice), Myerson, and many others. You can find Osborne and Rubinstein online here: http://books.osborne.economics.utoronto.ca/  
> **username_0**: Great, that's even better. I didn't know where to look. Thanks.  
> **username_2**: The definition of a game varies depending on the *kind* of game in question, e.g. extensive vs normal form, complete vs incomplete information, etc. Besides Osborne-Rubinstein, I'd also recommend Mas-Colell, Whinston, & Green (aka MWE).  

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## 1 answers

### Answer by username_3 on 2015-02-14. Score: 8 | Accepted answer

Your question is a little bit vague, but I'll do my best.

Your professor was motioning in the correct mathematical direction, but I think that what you're asking for is a more complete formalization.

Any game can be thought of as a tree (i.e. a non-cyclic graph). So to answer your first point about turn-taking, you might think about a function which maps nodes of the tree to some $i\in\{1,2,...n\}$ In the picture below, you might be able to define a formula to characterize the turn taking, but you can't necessarily do this in general. In this example $P\_1$ and $P\_2$ take alternating sequential turns. $P\_1$ can choose $L$ or $R$, then $P\_2$ can see which he chose, and then also choose $L$ or $R$.

![Game Tree Example](https://i.sstatic.net/4cNLP.gif)

Before going on to your second point, I need to take a moment to define information sets, which it doesn't sound like you've gotten to yet. Colloquially, an infoset is a set of vertices that represent uncertainty. The root of a game tree is always a singleton infoset, for example. Consider the game where you and I simultaneously choose heads or tails. When $P\_2$ chooses whether or not to play heads or tails, she doesn't know if I have chosen to play heads or tails either. Thus while the game tree will have 3 internal vertices (1 that is the root for $P\_1$, and 2 for $P\_2$, reflecting $P\_1$'s choice) there will only be 2 infosets because both vertices for $P\_2$ will be contained in the same infoset. In games where turn-taking is sequential, all infosets should be singleton.

Now back to your question about strategies. Each player has a set of strategies. In the example tree provided above $S\_1 = \{LLLLL,LLLLR,...,RRRRR\}$. This might seem a little bit confusing at first as it seems like each element of $S\_1$ goes above and beyond defining a path. Each player's strategy is defined in terms of infosets. Therefore, the element $LLLLL$ should be read as, when $P\_1$ finds herself at the top, she will choose $L$, and when she finds herself at the vertex that is farthest on the left and 1 down from the top, she will also choose $L$. It then also says that if $P\_2$ were to choose $R$ instead, that $P\_1$ would choose $L$. Thus a strategy is the set of all possible actions a player might take under all possible contingencies.

Finally, how do we state the mapping from strategies to payoffs? First we note that ever player has a strategy set $S\_i$, and then form the cartesian product $S=S\_1 \times S\_2 \times ... \times S\_n$, which characterizes all possible strategy combinations for all possible players. Then our payoff will be $\pi:S\to\mathbb{R}^n$. In our example $\pi(LLLLL,LLLLLLLLLL) = (4,2)$.


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