# Game theory in R with the new gtree package

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gtree is a new R package that allows to specify extensive form games using stages, similar as one specifies economic experiments with ztree or otree.

Let us specify a simple ultimatum game in gtree. A proposer can offer the responder between 0 and 10 Euros. If the responder accepts the offer, the proposer keeps 10 Euro minus the proposed amount. If the responder rejects, both get nothing.

```
library(gtree)
game = new_game(
gameId = "Ultimatum_Dictator",
options = make_game_options(verbose=FALSE),
params = list(numPlayers=2,cake=10),
stages = list(
stage("proposerStage",
player=1,
actions = list(
action("offer",~0:cake)
)
),
stage("responderStage",
player=2,
observe = c("offer"),
actions = list(
action("accept",c(FALSE,TRUE))
)
),
stage("payoffStage",
compute=list(
payoff_1 ~ ifelse(accept, cake-offer,0),
payoff_2 ~ ifelse(accept, offer,0)
)
)
)
)
```

The following code solves for all pure strategy subgame perfect equilibria and shows the outcomes.

```
game %>%
game_solve() %>%
eq_outcomes() %>%
select(eqo.ind,offer, accept, payoff_1, payoff_2)
```

```
## # A tibble: 2 x 5
## eqo.ind offer accept payoff_1 payoff_2
##
```
## 1 1 1 TRUE 9 1
## 2 2 0 TRUE 10 0

We have two different equilibrium outcomes: the proposer either offers 0 or 1 and in both equilibrium outcomes the offer will be accepted.

The function `eq_tables()`

also shows us the behavior off the equilibrium path:

```
game %>% eq_tables()
```

```
## $offer
## # A tibble: 2 x 2
## offer eq.inds
##
```
## 1 0 2
## 2 1 1
##
## $accept
## # A tibble: 3 x 2
## accept eq.inds
##
## 1 FALSE 1
## 2 TRUE 2
## 3 TRUE 1,2

We see that all offers above 0 are always accepted, and the two equilibria differ by whether an offer of 0 is accepted or rejected.

In behavioral economics the ultimatum game is often used to illustrate that many people have *social preferences* that go beyond the maximization of own payoffs only. One simple form of social preferences is inequality aversion (Fehr & Schmidt, 1999) The following code solves the game assuming inequality averse preferences with an envy parameter of `alpha=0.5`

but no guilt (`beta=0`

). This means the effective utility function of both players are given by

```
`u_1 = payoff_1 - 0.5*max(payoff_2-payoff_2,0)`
`u_2 = payoff_2 - 0.5*max(payoff_1-payoff_1,0)`
```

```
game %>%
game_set_preferences(pref_ineqAv(alpha=0.5, beta=0)) %>%
game_solve() %>%
eq_tables()
```

```
## $offer
## # A tibble: 1 x 2
## offer eq.inds
##
```
## 1 3 1
##
## $accept
## # A tibble: 11 x 3
## # Groups: offer [11]
## offer accept eq.inds
##
## 1 0 FALSE 1
## 2 1 FALSE 1
## 3 2 FALSE 1
## 4 3 TRUE 1
## 5 4 TRUE 1
## 6 5 TRUE 1
## 7 6 TRUE 1
## 8 7 TRUE 1
## 9 8 TRUE 1
## 10 9 TRUE 1
## 11 10 TRUE 1

Now we have a unique equilibrium in which the proposer offers 3 and every offer below 3 would be rejected.

The internal gtree solver used above is quite limited. It can only compute pure strategy subgame perfect equilibria. To access a much larger set of game theoretic solvers, one can use gtree together with the Gambit command line solvers. The following code uses the gambit-logit solver to compute a logit quantal response equilibrium (QRE) for the ultimatum game using the previously set inequality aversion preferences:

```
game %>%
game_gambit_solve(qre.lambda=2) %>%
eq_tables()
```

```
## $offer
## # A tibble: 9 x 3
## # Groups: offer [9]
## offer .prob eq.inds
##
```
## 1 0 0.00000116 1
## 2 1 0.00000120 1
## 3 2 0.00000718 1
## 4 3 0.602 1
## 5 4 0.353 1
## 6 5 0.0438 1
## 7 6 0.000646 1
## 8 7 0.00000952 1
## 9 8 0.000000140 1
##
## $accept
## # A tibble: 20 x 4
## # Groups: offer, accept [20]
## offer accept .prob eq.inds
##
## 1 0 FALSE 1.000 1
## 2 0 TRUE 0.0000264 1
## 3 1 FALSE 0.998 1
## 4 1 TRUE 0.00179 1
## 5 2 FALSE 0.892 1
## 6 2 TRUE 0.108 1
## 7 3 FALSE 0.108 1
## 8 3 TRUE 0.892 1
## 9 4 FALSE 0.00179 1
## 10 4 TRUE 0.998 1
## 11 5 FALSE 0.0000264 1
## 12 5 TRUE 1.000 1
## 13 6 FALSE 0.00000320 1
## 14 6 TRUE 1.000 1
## 15 7 FALSE 0.000000388 1
## 16 7 TRUE 1 1
## 17 8 FALSE 0.0000000471 1
## 18 8 TRUE 1 1
## 19 9 TRUE 1 1
## 20 10 TRUE 1 1

We see how player 1 now makes all offers with positive probabilities but mainly concentrates on offers 3,4 and 5. Interestingly, an offer of 2 is still very unlikely and will be rejected with very high probability.

Gambit has an own Python interface to directly construct the game tree of extensive form games. While it is a matter of taste, I think for many games studied in economic experiments it is simpler to specify stages instead of directly constructing a game tree. For a comparision you can look at a gtree implementation of a sender-receiver game whose Gambit Python interface implementation is linked as example on the Gambit main page.

Take a look at the gtree page https://skranz.github.io/gtree for more examples and features. On Github you can find gtree here: https://github.com/skranz/gtree. Please open an issue if you have some feature request, bug report or question concerning gtree.

## Playing games in the web: Rock Paper Scissors Laser Mirror

gtreeWebPlay is a companion package to gtree that helps building simply shiny apps that a allow to play a game. Below I embedded an app to play Rock-Paper-Scissors-Laser-Mirror:

Laser beats rock, paper and scissors.

Mirror beats laser but is beaten by rock, paper and scissors.

Otherwise as usual.

You play agains the population of all previous users. (And the first users play against the equilibrium strategies.) Let us see whether you can beat the crowd:

### Kuhn Poker

Below is a gtreeWebPlay app that allows you to play a round of Kuhn Poker.

(https://skranz.github.io/gtree/articles/t03_kuhn_poker.html). The deck has only three cards: *Jack*, *Queen* and *Ace* and two players who each get one card after they both put one dollar into the blind. Below is an embedded app that allows you to play it against a randomly chosen earlier player:

A game theoretic analysis of Kuhn poker using gtree can be found in this tutorial

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