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TreeGamePayoff
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TreeGamePayoff

New in 14.2

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TreeGamePayoff[tgame,strat]

gives the expected payoff for each player in the tree game tgame with strategy profile strat.

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TreeGamePayoff[tgame,strat,adv]

gives the expected payoff for each player in the tree game tgame with the incomplete strategy profile strat and the adversary type adv.

Details

  • TreeGamePayoff is also known as the expected payoff or the expected utility of a sequential game.
  • TreeGamePayoff is typically used to evaluate expected payoffs for players given strategies for each of the players.
  • A tree game strategy <|…,playeri<|{ni,1}pri,1,…,{ni,mi}pri,mi|>,…|> specifies the action probabilities pri,j for playeri in action node ni,j for playeri.
  • The expected payoff pi for player i at the action node nj with mj choices is given by pi[{nj}]=pi[{nj,k}].prj,k, while the expected payoff pi for player i at the end node nt is given by its specified payoff: pi[{nt}]=pi,j.
  • An incomplete tree strategy is a tree game strategy where only a subset of the action nodes have specified probabilities.
  • For an incomplete strategy, the following specifications can be given for unspecified actions adv:
  • "Perfect"the payoffs of a random trial assuming an optimal choice for unspecified actions
    "PerfectExpectation"the average payoffs assuming an optimal random choice for unspecified actions
    "Random"the payoffs of a random trial assuming a uniform random choice for unspecified actions
    "RandomExpectation"the average payoffs assuming a uniform random choice for unspecified actions

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Generate a three-player tree game:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-p22le
Out[1]=1

Find the expected payoffs for a given strategy:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-xxvrz
Out[2]=2

Payoffs for Revolution where players act optimally, using an empty association for a "perfect" strategy:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-f6v4ik
Out[1]=1

Payoffs for Beer-Quiche where "Weak" is chosen by the "Nature" player and all others act randomly:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-ezisz4
Out[1]=1

Scope  (3)Survey of the scope of standard use cases

Generate a two-player tree game:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-ix72io
Out[1]=1

Find the expected payoffs for a given strategy:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-cs84kh
Out[2]=2

Consider a two-player game with probabilities:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-sjtwz
Out[1]=1

Find the expected payoffs for a given strategy:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-9gzwl
Out[2]=2

Compute the expected payoffs for random players in a Centipede game:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-efyu0d
Out[1]=1

Applications  (5)Sample problems that can be solved with this function

Social Games  (3)

The Centipede game has two players who alternate in making decisions. At each turn, a player can choose between going "down" and ending the game or going "across" and continuing it (except at the last node where going "across" also ends the game). The longer the game goes on, the higher the total utility. A player who ends the game early will get a larger share of what utility there is. Consider a centipede game of three actions per player:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-togse
Out[1]=1

While the game-theoretic best solution is the pessimistic first choice, note that random players have a similar expected payoff. Compare the expected output of random and perfect players:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-b3xp92
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/08uokgdd4v8czxf-rqm4h
Out[3]=3

The tree game of Matching Pennies is a game where each of two people chooses either Head or Tail. If the choices differ, person 1 pays person 2 a dollar; if they are the same, person 2 pays person 1 a dollar. Generate a tree game of Matching Pennies:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-hdx5rm
Out[1]=1

Visualize the game:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-dhmfy0
Out[2]=2

Clearly, this game gives an advantage to player 2, making it impossible to get a better payoff against a perfect player. Imagine being player 1. Try several strategies to attempt to beat a perfect player (to have a payoff of 1):

In[3]:=3
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https://wolfram.com/xid/08uokgdd4v8czxf-en7elr
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/08uokgdd4v8czxf-j8fk6m
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/08uokgdd4v8czxf-eexkgt
Out[5]=5

An entry game is a game where an entrant decides whether to enter a market or not, and the incumbent decides whether to fight or accommodate the entrant if he enters. Generate an entry game:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-fsyxii
Out[1]=1

Show the decision tree:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-cdxj7y
Out[2]=2

Suppose that you are the incumbent player, and your strategy is to always fight (first choice). Find the expected payoff against a random player:

In[3]:=3
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https://wolfram.com/xid/08uokgdd4v8czxf-bwjqig
Out[3]=3

Perfect players assume perfect subgame equilibrium. Since that strategy is suboptimal, it turns out that the perfect player may have a worse payoff than the random player. Find the expected payoff against a perfect player:

In[4]:=4
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https://wolfram.com/xid/08uokgdd4v8czxf-erq3h
Out[4]=4

Recreational Games  (2)

Rock Paper Scissors is a zero-sum game, where either one player wins and the other loses, or there is a tie. Generate the tree version of this game, where the second player can choose an action considering the action of the first player:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-ubinn
Out[1]=1

Visualize the game:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-dyl7at
Out[2]=2

Clearly, in the case of the tree game of Rock Paper Scissors, the second player can always choose the best action, and thus the first player is at a disadvantage by playing first. Write the perfect strategy in this version of Rock Paper Scissors:

In[3]:=3
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https://wolfram.com/xid/08uokgdd4v8czxf-dbk6o
Out[3]=3

Consider the game of tic-tac-toe. Design a function to generate the partial tic-tac-toe game between players A and B:

In[1]:=1
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https://wolfram.com/xid/08uokgdd4v8czxf-dd9gx5

Consider a partially completed game of tic-tac-toe:

In[2]:=2
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https://wolfram.com/xid/08uokgdd4v8czxf-dw968s

Form a TreeGame based on it:

In[3]:=3
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https://wolfram.com/xid/08uokgdd4v8czxf-by41zt
Out[3]=3

If both players play perfectly, Player B wins:

In[4]:=4
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https://wolfram.com/xid/08uokgdd4v8czxf-cmntih
Out[4]=4

If both players play randomly, player 2 is more likely to win only by a small margin:

In[5]:=5
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https://wolfram.com/xid/08uokgdd4v8czxf-bhypnq
Out[5]=5
Wolfram Research (2025), TreeGamePayoff, Wolfram Language function, https://reference.wolfram.com/language/ref/TreeGamePayoff.html.
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Wolfram Research (2025), TreeGamePayoff, Wolfram Language function, https://reference.wolfram.com/language/ref/TreeGamePayoff.html.

Text

Wolfram Research (2025), TreeGamePayoff, Wolfram Language function, https://reference.wolfram.com/language/ref/TreeGamePayoff.html.

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Wolfram Research (2025), TreeGamePayoff, Wolfram Language function, https://reference.wolfram.com/language/ref/TreeGamePayoff.html.

CMS

Wolfram Language. 2025. "TreeGamePayoff." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TreeGamePayoff.html.

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Wolfram Language. 2025. "TreeGamePayoff." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TreeGamePayoff.html.

APA

Wolfram Language. (2025). TreeGamePayoff. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TreeGamePayoff.html

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Wolfram Language. (2025). TreeGamePayoff. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TreeGamePayoff.html

BibTeX

@misc{reference.wolfram_2025_treegamepayoff, author="Wolfram Research", title="{TreeGamePayoff}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/TreeGamePayoff.html}", note=[Accessed: 16-April-2025 ]}

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@misc{reference.wolfram_2025_treegamepayoff, author="Wolfram Research", title="{TreeGamePayoff}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/TreeGamePayoff.html}", note=[Accessed: 16-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_treegamepayoff, organization={Wolfram Research}, title={TreeGamePayoff}, year={2025}, url={https://reference.wolfram.com/language/ref/TreeGamePayoff.html}, note=[Accessed: 16-April-2025 ]}

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@online{reference.wolfram_2025_treegamepayoff, organization={Wolfram Research}, title={TreeGamePayoff}, year={2025}, url={https://reference.wolfram.com/language/ref/TreeGamePayoff.html}, note=[Accessed: 16-April-2025 ]}