game theory - A formal language for the representation and analysis of interactive situations where several players take actions that jointly determine the final outcome. [GT2018:p11] [GT2018:p18]
player - An agent that acts purposefully and logically to get as close as possible to a set of well-defined goals. What a player is depends on the context. For example, in the context of ...
homo rationalis - An agent that acts purposefully and logically to get as close as possible to a set of well-defined goals. [GT2018:p12]
cooperative game theory - A branch of game theory where players can cooperate through communication, forming coalitions, and making binding agreements. An example of cooperative game theory is politics, where various groups often form larger coalitions. [GT2018:p12]
non-cooperative game theory - A branch of game theory where players are either unable to communicate or unable to make binding agreements. An example of non-cooperative game theory is competing firms that are hampered from colluding due to anti-trust laws. [GT2018:p12]
ordinal payoff - A game where the preferences of discrete outcomes are ranked using numbers. For example, the preference for the type of cake to consume at a birthday party may be ...
Item 3 (chocolate cake) is preferable over item 2 (non-chocolate cake), which is preferable over 1 (no cake). [GT2018:p12]
cardinal payoff - A game where the preference of outcome is based on quantities (e.g. amount of money) rather than a ranking of discrete items based on preference (ordinal payoffs). [GT2018:p12]
selfish and greedy - A player that focuses exclusively on itself when evaluating outcomes (selfish) and has preference for an outcome where it gains more over less (greedy). [GT2018:p18]
fair-minded - A player whose preference is that all players benefit equally. [GT2018:p18]
benevolent - A player that, when comparing a set of possible outcomes where they gain nothing, has a preference for other players gaining more over less. [GT2018:p18]
preference - Given a set of outcomes, a player's preferences are the order in which those outcomes are desired. It's not safe to assume that all players in a game are selfish and greedy. For example, a player may benevolent or fair-minded. [GT2018:p18]
There are four common ways of representing preferences. Imagine a scenario where, given the outcomes {o1, o2, o3, o4}, the preference for those outcomes is o3 is better than o1, which is just as good as o4, which in turn is better than o2 (best is o3, worst is o2).
Relational notation - Represent which outcome has preference over the other using ...
>: A > B means outcome A is better than outcome B.==: A == B means outcome A is just as good as outcome B.These preferences are typically written chained together (e.g. A > B == C is equivalent to A > B and B == C). [GT2018:p20]
The scenario above would be represented as o3 > o1 == o4 > o2.
⚠️NOTE️️️⚠️
The book also provides the following notation: >=, where A >= B means outcome A is at least as good as outcome B. Then it goes through a convoluted section that explains > and == can be represented just by >=. For example, given two outcomes {A, B}, the relation ...
A == B can be represented using just >= as A >= B and B >= A.A > B can be represented using just >= as A >= B and not(B >= A).The problem is, I don't know what "at least as good as" means in terms of the desire for an outcome. If something A is "at least as good as" B (e.g. A >= B), there's the possibility that I'll gain more from A vs B, so why wouldn't I always prefer A over B?
Cartesian product - Given the cartesian product of the outcomes with itself, keep only the ordered pairs [A, B] where A is at least desired as much as B. [GT2018:p20]
The scenario above would be represented as follows.
[
[o1, o1], [o1, o4], [o1, o2],
[o2, o2],
[o3, o3], [o3, o1], [o3, o4], [o3, o2],
[o4, o4], [o4, o1], [o4, o2]
]
Ranked list - Represent which outcome has preference as a list from best-to-worst, where equally desired outcomes are put into the same index. [GT2018:p21]
The scenario above would be represented as follows.
Number assignment - Represent which output has preference by assigning a number to each outcome, where larger numbers are desired more than smaller numbers. [GT2018:p21]
The scenario above could be represented as the following mapping {o3=30, o1=19, o4=19, o2=6}.
⚠️NOTE️️️⚠️
Assigning numbers in this way is sometimes referred to as a utility function. [GT2018:p21]
utility function - Given a set of outcomes and preferences for those outcomes, a utility function assigns a number to each outcome, where that number identifies that outcome's ranking in terms of preference. Given two outcomes A and B, if ...
A>B), A is assigned a higher number than B.A==B), A is assigned the same number as B.The numbers are simply identifying ranking based on best-to-worst, they aren't quantities. [GT2018:p21] As such, there are an infinity number of utility functions that represent the same preference. [GT2018:p22] For example, o3 > o1 == o4 > o2 may be mapped by a utility function as {o3=30, o1=19, o4=19, o2=6}, {o3=3, o1=2, o4=2, o2=-50}, {o3=70, o1=3, o4=3, o2=1}, etc..
⚠️NOTE️️️⚠️
If the outcomes are drink Coke vs drink Sprite and the utility function assigns `{Coke=30, Sprite=10}, that doesn't mean Coke is 3 times more as desired as Sprite. It simply means Coke is preferred over Sprite.
strategy - One of many choices that a player has in a game. [GT2018:p19] For example, Alice and Bob are the players of a game where ...
[Lie, Get Violent, Stay Silent].[Truth, Lie].strategy profile - Given a set of players and their strategies [[P1, S1], [P2, S2], ..., [Pn, Sn]], S is the cartesian product of each player's strategies: S1 x S2 x ... Sn. Each element in S is called a strategy profile. [GT2018:p19]
For example, Alice and Bob are the players of a game where ...
[Lie, Get Violent, Stay Silent].[Truth, Lie].The cartesian product of these strategies are listed below. Each entry is a strategy profile.
[Lie, Truth][Lie, Lie][Get Violent, Truth][Get Violent, Lie][Stay Silent, Truth][Stay Silent, Lie]game frame - The cartesian product of each player's strategies in a game, where each output has a set of outcomes. [GT2018:p18-19] For example, the game-frame below is for a scenario where Alice's strategies are [Lie, Get Violent, Stay Silent] and Bob's strategies are [Truth, Lie].
[Lie, Truth]: A freed, B jailed
[Lie, Lie]: A freed, B freed
[Get Violent, Truth]: A jailed, B jailed
[Get Violent, Lie]: A jailed, B freed
[Stay Silent, Truth]: A freed, B jailed
[Stay Silent, Lie]: A freed, B freed
If there are only two players, a game-frame can be represented as a table. [GT2018:p19]
| Truth | Lie | |
|---|---|---|
| Lie | A freed, B jailed | A freed, B freed |
| Get Violent | A jailed, B jailed | A jailed, B freed |
| Stay Silent | A freed, B jailed | A freed, B freed |
game frame in strategic form - Representation of a game frame as the tuple [I, S, O, f], where ...
I is the set of players: [1, 2, ..., n]S is the set of strategies each player has: [S1, S2, ..., Sn]O is the set of outcomes: [O1, O2, ..., Om].f is a function that maps each strategy profile in S to an outcome in O.⚠️NOTE️️️⚠️
Each index in S maps to the player at the same index in I. For example, S2 is the set of strategies for player 2.
S on its own is represented as the complete set of strategy profiles for the game: The cartesian product of [S1, S2, ..., S4]. For each strategy profile s in S, f(s) E O.
game - A game frame along with the preferences of each player in that game frame. [GT2018:p19]
For example, given players A and B with the following game frame, ...
| B: Truth | B: Lie | |
|---|---|---|
| A: Lie | A freed / B jailed | A freed / B freed |
| A: Get Violent | A jailed / B jailed | A jailed / B freed |
| A: Stay Silent | A freed / B jailed | A freed / B freed |
The preference of player...
[A freed / B jailed, A freed / B freed, A jailed / B jailed, A jailed / B freed].[A freed / B freed, A jailed / B freed, A jailed / B jailed, A freed / B jailed].ordinal game in strategic form - Representation of an ordinal game as a game frame in strategic form with an extra item appended to the tuple: The preference for each player in the game. As such, the items of the tuple are as follows:
I is the set of players: [1, 2, ..., n]S is the set of strategies each player has: [S1, S2, ..., Sn] (S on its own is represented as the complete set of strategy profiles for the game).O is the set of outcomes: [O1, O2, ..., Om].f is a function that maps each strategy profile in S to an outcome in O.P is the set containing each player's preference rankings: [P1, P2, ..., Pn].⚠️NOTE️️️⚠️
Similar to S, each index in P maps to the player at the same index in I. For example, P2 is the preference ranking for player 2.
payoff function - Maps each strategy profile in a game to a player's preferences via a utility function. [GT2018:p22] For example, the game-frame below is for a scenario where Alice's strategies are {Lie, Get Violent, Stay Silent} and Bob's strategies are {Truth, Lie}.
| B: Truth | B: Lie | |
|---|---|---|
| A: Lie | A freed / B jailed | A freed / B freed |
| A: Get Violent | A jailed / B jailed | A jailed / B freed |
| A: Stay Silent | A freed / B jailed | A freed / B freed |
Listed from most-to-least desired, the preferences of player...
[A freed / B jailed, A freed / B freed, A jailed / B jailed, A jailed / B freed].[A freed / B freed, A jailed / B freed, A jailed / B jailed, A freed / B jailed].Once each player's preferences is passed through its respective utility function, those preferences are represented as numeric rankings. The numeric rankings for ...
{A freed / B jailed: 4, A freed / B freed: 3, A jailed / B jailed: 2, A jailed / B freed: 2}.{A freed / B jailed: 1, A freed / B freed: 10, A jailed / B jailed: 3, A jailed / B freed: 6}.Instead of mapping each outcome to a numeric rank (as is done above), a payoff function maps the strategy profile that leads to that outcome to a numeric rank. The payoff functions for Alice and Bob perform mappings as follows.
Alice:
- [Lie, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Lie, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
- [Get Violent, Truth]: 2 # outcome is A jailed / B jailed, which is ranked as 2
- [Get Violent, Lie]: 2 # outcome is A jailed / B freed, which is ranked as 2
- [Stay Silent, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Stay Silent, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
Bob:
- [Lie, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Lie, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
- [Get Violent, Truth]: 3 # outcome is A jailed / B jailed, which is ranked as 3
- [Get Violent, Lie]: 6 # outcome is A jailed / B freed, which is ranked as 6
- [Stay Silent, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Stay Silent, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
reduced-form ordinal game in strategic form - An ordinal game in strategic form but condensed by removing / replacing certain items in the tuple.
The following items have been removed from the tuple:
O is the set of outcomes: [O1, O2, ..., Om].f is a function that maps each strategy profile in S to an outcome in O.P is the set of preference each player has: [P1, P2, ..., Pn].The following items have been added to the tuple:
fp is the set containing each player's payoff function: [fp1, fp2, ..., fpn]As such, the items of the tuple are as follows:
I is the set of players: [1, 2, ..., n]S is the set of strategies each player has: [S1, S2, ..., Sn] (S on its own is represented as the complete set of strategy profiles for the game).fp is the set containing each player's payoff function: [fp1, fp2, ..., fpn]⚠️NOTE️️️⚠️
Similar to S, each index in fp maps to the player at the same index in I. For example, fp2 is the preference ranking for player 2.
Previously, player n had its preferences for outcomes O specified in Pn. The idea with this change is that, since a strategy profile in S leads to an outcome in O, player n can specify its preference by mapping strategy profiles directly to numeric rankings via fpn(). [GT2018:p22]
⚠️NOTE️️️⚠️
It's called reduced-form because some information has been lost. Specifically, O is no longer there, which actually spelled out what the outcomes were.
A table is commonly used to represent a reduced-form ordinal game in strategic form when there are only 2 players. [GT2018:p23] For example, Alice and Bob are the players in a game. The strategies for ...
[Lie, Get Violent, Stay Silent].[Truth, Lie].Once each strategy profile is passed through both Alice's payoff function and Bob's payoff functions, their preferences are encoded using the following numeric rankings:
Alice:
- [Lie, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Lie, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
- [Get Violent, Truth]: 2 # outcome is A jailed / B jailed, which is ranked as 2
- [Get Violent, Lie]: 2 # outcome is A jailed / B freed, which is ranked as 2
- [Stay Silent, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Stay Silent, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
Bob:
- [Lie, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Lie, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
- [Get Violent, Truth]: 3 # outcome is A jailed / B jailed, which is ranked as 3
- [Get Violent, Lie]: 6 # outcome is A jailed / B freed, which is ranked as 6
- [Stay Silent, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Stay Silent, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
The table lists out Alice's strategies on the left and Bob's strategies on the top. Each cell in the table maps to a strategy profile. Alice and Bob's numeric rankings for each strategy profile are paired together and placed into the cell for that strategy profile.
| B: Truth | B: Lie | |
|---|---|---|
| A: Lie | [4, 1] | [3, 10] |
| A: Get Violent | [2, 3] | [2, 6] |
| A: Stay Silent | [4, 1] | [3, 10] |
strict dominance - Given two strategies [A, B] for a player in a game, A is said to strictly dominate B if the strategy profiles containing A all have a higher payoff than the corresponding strategy profiles containing B. In this case, corresponding means that the other players's strategies must match between the strategy profiles. For example, the following two strategy profiles use the same strategies for all players other than the player at index 1: [Walk, Lie, Run, Cheat] vs [Walk, Truth, Run, Cheat]. [GT2018:p24]
gt_ch1_code/ReducedFormOrdinalGame.py (lines 26 to 36):
def strictly_dominates_other(self, player: int, strategy1: str, strategy2: str) -> bool:
for player_strategy_profile in product(*(s if i != player else {'FAKE'} for i, s in enumerate(self.strategies))):
player_strategy_profile_1 = list(player_strategy_profile)
player_strategy_profile_2 = list(player_strategy_profile)
player_strategy_profile_1[player] = strategy1
player_strategy_profile_2[player] = strategy2
payoff_1 = self.player_strategy_profile_preference(player, tuple(player_strategy_profile_1))
payoff_2 = self.player_strategy_profile_preference(player, tuple(player_strategy_profile_2))
if not (payoff_1 > payoff_2):
return False
return True
For example, Alice and Bob are the players in a game. The strategies for ...
[Lie, Get Violent, Stay Silent].[Truth, Lie].Once each strategy profile is passed through both Alice's payoff function and Bob's payoff functions, their preferences are encoded using the following payoffs:
Alice:
- [Lie, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Lie, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
- [Get Violent, Truth]: 2 # outcome is A jailed / B jailed, which is ranked as 2
- [Get Violent, Lie]: 2 # outcome is A jailed / B freed, which is ranked as 2
- [Stay Silent, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Stay Silent, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
Bob:
- [Lie, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Lie, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
- [Get Violent, Truth]: 3 # outcome is A jailed / B jailed, which is ranked as 3
- [Get Violent, Lie]: 6 # outcome is A jailed / B freed, which is ranked as 6
- [Stay Silent, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Stay Silent, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
⚠️NOTE️️️⚠️
It's easier to think of this in terms of a reduced game in strategic-form table. For example, the game above is represented as follows.
| B: Truth | B: Lie | |
|---|---|---|
| A: Lie | [4, 1] | [3, 10] |
| A: Get Violent | [2, 3] | [2, 6] |
| A: Stay Silent | [4, 1] | [3, 10] |
For Alice, each Lie payoff is greater than the corresponding Get Violent payoff (4>2 and 3>2), as such Lie strictly dominates Get Violent.
weak dominance - Given two strategies [A, B] for a player in a game, A is said to weakly dominate B if both of the following conditions are met:
A have a greater than or equal payoff than the corresponding strategy profiles containing B.A has a greater payoff than the corresponding strategy profile containing B.In this case, corresponding means that the other players's strategies must match between the strategy profiles. For example, the following two strategy profiles use the same strategies for all players other than the player at index 1: [Walk, Lie, Run, Cheat] vs [Walk, Truth, Run, Cheat]. [GT2018:p24]
⚠️NOTE️️️⚠️
If A strictly dominates B, it also meets the conditions to weakly dominate B (but not necessarily the other way around). When the book says weakly dominates, it usually means it weakly dominates but not strictly dominates. [GT2018:p26]
gt_ch1_code/ReducedFormOrdinalGame.py (lines 40 to 54):
def weakly_dominates_other(self, player: int, strategy1: str, strategy2: str) -> bool:
strict_found = False
for player_strategy_profile in product(*(s if i != player else {'FAKE'} for i, s in enumerate(self.strategies))):
player_strategy_profile_1 = list(player_strategy_profile)
player_strategy_profile_2 = list(player_strategy_profile)
player_strategy_profile_1[player] = strategy1
player_strategy_profile_2[player] = strategy2
payoff_1 = self.player_strategy_profile_preference(player, tuple(player_strategy_profile_1))
payoff_2 = self.player_strategy_profile_preference(player, tuple(player_strategy_profile_2))
if not (payoff_1 >= payoff_2):
return False
if payoff_1 > payoff_2:
strict_found = True
return strict_found
For example, Alice and Bob are the players in a game. The strategies for ...
[Lie, Get Violent, Stay Silent].[Truth, Lie].Once each strategy profile is passed through both Alice's payoff function and Bob's payoff functions, their preferences are encoded using the following payoffs:
Alice:
- [Lie, Truth]: 5 # outcome is A freed / B jailed, which is ranked as 4
- [Lie, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
- [Get Violent, Truth]: 2 # outcome is A jailed / B jailed, which is ranked as 2
- [Get Violent, Lie]: 2 # outcome is A jailed / B freed, which is ranked as 2
- [Stay Silent, Truth]: 4 # outcome is A freed / B jailed, which is ranked as 4
- [Stay Silent, Lie]: 3 # outcome is A freed / B freed, which is ranked as 3
Bob:
- [Lie, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Lie, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
- [Get Violent, Truth]: 3 # outcome is A jailed / B jailed, which is ranked as 3
- [Get Violent, Lie]: 6 # outcome is A jailed / B freed, which is ranked as 6
- [Stay Silent, Truth]: 1 # outcome is A freed / B jailed, which is ranked as 1
- [Stay Silent, Lie]: 10 # outcome is A freed / B freed, which is ranked as 10
strictly dominant - Given strategy A for a player in a game, A is said to be strictly dominant if it's strictly dominates every other strategy that player has. A strictly dominant strategy is colloquially known as the best strategy. [GT2018:p26]
gt_ch1_code/ReducedFormOrdinalGame.py (lines 71 to 77):
def strictly_dominates_overall(self, player: int, strategy: str) -> bool:
for other_strategy in self.strategies[player]:
if strategy == other_strategy:
continue
if not self.strictly_dominates_other(player, strategy, other_strategy):
return False
return True
weakly dominant - Given strategy A for a player in a game, A is said to be weakly dominant if, for every other strategy that player has, A is either equivalent or A weakly dominates it. [GT2018:p26]
gt_ch1_code/ReducedFormOrdinalGame.py (lines 81 to 88):
def weakly_dominates_overall(self, player: int, strategy: str) -> bool:
for other_strategy in self.strategies[player]:
if strategy == other_strategy:
continue
if not self.weakly_dominates_other(player, strategy, other_strategy) \
and not self.equivalent_to_other(player, strategy, other_strategy):
return False
return True
By definition, if a strategy is strictly dominant, it's also weakly dominant. To disambiguate, in most cases weakly dominant is interpreted as "weakly dominant but not strictly dominant". [GT2018:p27]
strictly dominant-strategy equilibrium - A strategy profile where the strategy for each player is strictly dominant. [GT2018:p27]
gt_ch1_code/ReducedFormOrdinalGame.py (lines 92 to 95):
def strictly_dominant_strategy_equilibrium(self, strategy_profile: list[str]):
for i, _ in enumerate(self.players):
if not self.strictly_dominates_overall(i, strategy_profile[i]):
return False
return True
weakly dominant-strategy equilibrium - A strategy profile where the strategy ...
gt_ch1_code/ReducedFormOrdinalGame.py (lines 100 to 106):
def weakly_dominant_strategy_equilibrium(self, strategy_profile: list[str]):
found_strict_dom = False
for i, _ in enumerate(self.players):
if not self.weakly_dominates_overall(i, strategy_profile[i]):
return False
found_strict_dom = self.strictly_dominates_overall(i, strategy_profile[i])
return found_strict_dom
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
TODO: pg28 top of page, make code example for prisoner dilemma
Recall game frame in strategic form: [I, S, O, f], where ...
I is the set of players: [1, 2, ..., n]S is the set of strategies each player has: [S1, S2, ..., Sn]O is the set of outcomes: [O1, O2, ..., Om].f is a function that maps each strategy profile in S to an outcome in O.As a game frame in strategic form, the game frame is described as ...
I: 2 # 2 players, 1 is Antonia and 2 is Bob
S:
- [Pick 2, Pick 4, Pick 6] # Player 1's strategies (Antonia)
- [Pick 1, Pick 3, Pick 5] # Player 2's strategies (Bob)
O: [Mexican, Italian, Japanese]
f:
[Pick 2, Pick 1]: Mexican
[Pick 2, Pick 3]: Mexican
[Pick 2, Pick 5]: Italian
[Pick 4, Pick 1]: Mexican
[Pick 4, Pick 3]: Italian
[Pick 4, Pick 5]: Japanese
[Pick 6, Pick 1]: Italian
[Pick 6, Pick 3]: Japanese
[Pick 6, Pick 5]: Japanese
As a table, the game frame is described as ...
| B: Pick 1 | B: Pick 3 | B: Pick 5 | |
|---|---|---|---|
| A: Pick 2 | Mexican | Mexican | Italian |
| A: Pick 4 | Mexican | Italian | Japanese |
| A: Pick 6 | Italian | Japanese | Japanese |
Recall reduced-form ordinal game in strategic form: [I, S, fp], where ...
I is the set of players: [1, 2, ..., n]S is the set of strategies each player has: [S1, S2, ..., Sn] (S on its own is represented as the complete set of strategy profiles for the game).fp is the set containing each player's payoff function: [fp1, fp2, ..., fpn]As a reduced-form ordinal game in strategic form, the game is described as ...
I: 2 # 2 players, 1 is Antonia and 2 is Bob
S:
- [Pick 2, Pick 4, Pick 6] # Player 1's strategies (Antonia)
- [Pick 1, Pick 3, Pick 5] # Player 2's strategies (Bob)
fp:
- # Antonia's strategies to preference rankings
[Pick 2, Pick 1]: 3 # Mexican
[Pick 2, Pick 3]: 3 # Mexican
[Pick 2, Pick 5]: 2 # Italian
[Pick 4, Pick 1]: 3 # Mexican
[Pick 4, Pick 3]: 2 # Italian
[Pick 4, Pick 5]: 1 # Japanese
[Pick 6, Pick 1]: 2 # Italian
[Pick 6, Pick 3]: 1 # Japanese
[Pick 6, Pick 5]: 1 # Japanese
# Bob's strategies to preference rankings
- [Pick 2, Pick 1]: 2 # Mexican
[Pick 2, Pick 3]: 2 # Mexican
[Pick 2, Pick 5]: 3 # Italian
[Pick 4, Pick 1]: 2 # Mexican
[Pick 4, Pick 3]: 3 # Italian
[Pick 4, Pick 5]: 1 # Japanese
[Pick 6, Pick 1]: 3 # Italian
[Pick 6, Pick 3]: 1 # Japanese
[Pick 6, Pick 5]: 1 # Japanese
As a table, the reduced-form ordinal game in strategic form is described as ...
| B: Pick 1 | B: Pick 3 | B: Pick 5 | |
|---|---|---|---|
| A: Pick 2 | [3,2] |
[3,2] |
[2,3] |
| A: Pick 4 | [3,2] |
[2,3] |
[1,1] |
| A: Pick 6 | [2,3] |
[1,1] |
[1,1] |
As a table, the game frame is described as ...
| B: Pick 0 | B: Pick 1 | B: Pick 2 | |
|---|---|---|---|
| A: Pick 2 | A:0/B:+2 | A:0/B:+3 | A:0/B:+2 |
| A: Pick 4 | A:0/B:+2 | A:+5/B:+5 | A:+4/B:+2 |
| A: Pick 6 | A:+4/B:+2 | A:+3/B:+7 | A:+2/B:0 |
Recall that a player being selfish and greedy means that it only cares about how much it gain and prefers more gain over less
If both Antonia and Bob are selfish and greedy, as a table, the reduced-form ordinal game in strategic form is described as ...
| B: Pick 0 | B: Pick 1 | B: Pick 2 | |
|---|---|---|---|
| A: Pick 2 | [0,2] |
[0,3] |
[0,2] |
| A: Pick 4 | [0,2] |
[5,5] |
[4,2] |
| A: Pick 6 | [4,2] |
[3,7] |
[2,0] |
| C: Yes Press | C: No Press | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
| C: Yes Press | C: No Press | ||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
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| C: Yes Press | C: No Press | ||||||||||||||||||
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