Analysis for design

Analysis for Design
Evaluating game system behavior
Petri Lankoski Södertörn University

How Is Your Probability Math?
Dice
 Probability of 1d6 to get 6?
 1/6
 Probability of 1d6 to get 6 after
previous was 6?
 1/6
 Probability of 2d6 to get double
6?
 1/36
 Probability of 2d6 sum is 7?
 6/36 (6 cases where the sum is
7 out of 36 possibilities)
Cards
 Probability to draw an ace?
 4/52
 Probability to draw second ace after
an ace?
 3/51
 Probability to draw an ace if the first
was not an ace?
 4/51
 Draw a card and put it face down.
What is the probability the the second
card is an ace?
 4/52*3/51 (both aces) + 48/52*4/51
(first not ace, second is ace)

Sum of Dice, 2d6
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
Sum Prob
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36
3 fours

Dice vs Cards
 History does not influence probabilities of dice
 Drawing a card influence probabilities of next
drawn card
 Shuffling resets
 Very handy to get certain kind of results
 Catan: all boards have 4 hex producing wood
and 3 producing stone and so on
 Both can be simulated using pseudo-random
numbers

More about randomness
 Salen & Zimmerman, 2004. Rules of play.
 Game as systems of uncertainty, pp.173–188.
 Elias, Garfield & Gutschera, 2012. Characteristics
of games.
 Indeterminacy, pp. 137–166

A Skill System
skilld10 vs difficulty example

Skill check using Xd10
 Throw skilld10
 Success if at least one die over difficulty
 If throw is 1 reduce one success
 If negative amount of success skill check is
fumble
 Is this good?

10000
20000
30000
40000
50000
2 4 6 8
diff
success
skill
1
2
3
4
5
6
7
8
9
10
0
10000
20000
30000
2 4 6 8
diff
fail
skill
1
2
3
4
5
6
7
8
9
10
0
2500
5000
7500
10000
2 4 6 8
diff
fumble
skill
1
2
3
4
5
6
7
8
9
10

What we learned
 System works OK in most cases
 Skill level 1 and 2 differently to other levels
 High difficulties are significantly less likely to fumble
with skill level 1 than with high skills; skill level 2 less
anomaly, but still very different
 Skill levels 1 and 2 are less likely to success in most
cases
 But behaves differently than other skill levels
 One possible fix:
 No skill rolls use 1 or 2 dice

Trouble
A simple example
Image: Wikipedia

Trouble
 How long (rounds & minutes ) in average game
takes?
 Ignore returning home by landing on them, on
board with with 6 and home with only correct roll

Answer: Trouble
 Expected value: long run average
 D6: expected value = 3.5 (1+6/2), but…
 However, the Trouble die expected value is 4
 Simulated the die 50000 times and calculated
average
 Track length: 32, 31, 30, 29
 Game time estimate (underestimates)
 Average rounds to complete
 32 / 4 + 31 / 4 + 30 / 4 + 29 / 4 = 30.5 rounds
 Turn: 15 sec -> round: 1 min -> game: 30.5 * 1 min ~30
min
 But return home rule makes game more unpredictable

Balance
 Is game balanced?
 How game is balanced?

Answer: Balance
 Game is symmetrical
⇒balanced
 Except:
 1st (and so on) player has a small advantage over
next ones
 However, amount of rounds and return to home
mechanism is likely to reduce the advantage

Assignment
Island Tour

Game
Rules
 D6 to move
 Winner is the one who visit all
listed squares first (see the
column on the right hand
side)
 Must stop to a site
 All players start at START
 In one lands to red place,
one looses ones next turn
 Board in next slide
Goals
 Player 1 visits
 1, 6, 9, 15, 20, 32
 Player 2 visits
 4, 10, 13, 18, 27, 30
 Player 3 visits
 7, 8, 11, 15, 22, 29
 Player 4 visits
 5, 10, 11, 21, 23, 26
Petri Lankoski Södertörn University Assignment is based on Korkeasaari board game

Board
Letters are marking junctions

Island Tour
 Estimate how much time game takes
 How would you balance the game if it is not
balanced?
 Do not change core mechanics
 Racing with die
 Asymmetric

Answers: Island Tour
Length Wait a
round
squares
Obl. wait
a round
squares
Expected
time
(rounds)
Player 1 69 5 1 20.8
Player 2 72 3 0 20.6
Player 3 79 7 0 22.7
Player 4 70 4 0 20.1
Estimated time = length / 3.5 + wait a round / length + obl. wait a round
Probability to land

Answers: Island Tour
 Player 3 has longer / harder path
 Easy fix: shorten the path
 E.g. 7, 8, 15, 22, 28, 31

Simulation
 How the players gain resources
 Simplified
 Robber vs no robber discard
 Only resource amount simulated, not types
 Assumptions
 Four player game
 0-3 resources at hand when ones turn starts
 Model for using resources; not able to use all resources
 Better robber simulation
 One specific board set-up
 The results does not vary much board to board
 The results can vary with not optimal settlement placements
 50 000 iterations used

Simulation set-up
• 2 victory point set-up

Model
#!/usr/bin/python
import random
from collections import Counter
# board model (2 victory points)
field1 = {
2: {'white': 0, 'blue':0, 'red': 0, 'orange': 0},
12: {'white': 0, 'blue':0, 'red': 0, 'orange': 0}
}
 The above model does not contain handling for
robber
 The code for simulating this model is bit more
complicated

Startposition
comparison
Petri Lankoski Södertörn University Dark blue: 50%, light blue: 80%
4
5 5
6
0 0
1 11
2 2
3
3 3
4
5
2.05356
2.60712
3.16662
3.72224
0
2
4
6
0 1 2 3
Turn
Resources White
4
5
6
7
0 0
1 11
2 2
3
3
4 4
5
2.08214
2.6593
3.2414
3.81416
0
2
4
6
0 1 2 3
Turn
Resources
Blue
4
5 5
6
0 0
1
2
1
2 2
3
3
4 4
5
2.08276
2.66588
3.24844
3.83248
0
2
4
6
0 1 2 3
Turn
Resources
Orange
4
5 5
6
0 0
1
2
1
2 2
3
3
4 4
5
2.07952
2.66072
3.2421
3.82596
0
2
4
6
0 1 2 3
Turn
Resources Red

What this mean?

What this mean?
 Rather well-balanced starting positions
 No advantage/disadvantage for any color
 White slightly lower average resource gain
 But have a port
 Blue slightly have more variation in resource gain
(80% area is wider)

Impactofsetup

Impactofsetup
3
4
5 5
0 0
1 11 1
2 2
3 3
4 4
1.89398
2.28342
2.67204
3.06254
0
2
4
6
0 1 2 3
Turn
Resources
White, bad
3
4
5 5
0 0 0 0
1 1 1
2
3 3 3
4
1.8087
2.11382
2.42424
2.72642
0
2
4
6
0 1 2 3
Turn
Resources
White, bad alt 2
4
5 5
6
0 0
1
2
1
2 2
3
3
4 4
5
2.08718
2.67596
3.26016
3.84286
0
2
4
6
0 1 2 3
Turn
Resources
White, good alt
4
5 5
6
0 0
1 11
2 2
3
3 3
4
5
2.05356
2.60712
3.16662
3.72224
0
2
4
6
0 1 2 3
Turn
Resources
White

What this mean?
 Initial placement of ones settlement is important
 Rather big impact on resource gain
 Even bigger after upgrading settlements to cities

Feedback loop & Roll 7
Effect?
Scenario
• Only white
simulated
• White built cities
in all places
marked

Feedbackloop&roll
“7”effect
6
11
12
14
1 1
2
3
2
3
4
64
6
8
10
3.28772
5.07054
6.68098
8.11302
0
5
10
15
0 1 2 3
Turn
Resources Robber=Default
6
11
13
15
1 1
2
3
2
3
4
64
6
8
11
3.2899
5.0725
6.85024
8.62992
0
5
10
15
0 1 2 3
Turn
Resources
Robber=No
6
11
12
14
1 1
2 22
3
4
5
4
6
8
10
3.27528
5.04646
6.527
7.68936
0
5
10
15
0 1 2 3
Turn
Resources
Robber=To zero
6
11
12
14
1 1
2
3
2
3
4
5
4
6
8
10
3.29602
5.0909
6.50472
7.7635
0
5
10
15
0 1 2 3
Turn
Resources Robber=Limit 4 & half

What this mean?
 Feedback loop is weakened by the board design
 There is no equally good places to build after initial
setup
 Robber (rolling 7) makes lucky streaks rarer
 Not big effect on positive feedback look on average
 What if scenarios
 Robber -> discard all
 Discard if more than four resources

Monopoly

Board & Movement
Chance to end
Up In a square
1/40 = 2,50%?
3 doubles
in a row
1/16 Card takes
to Jail
A player can
increase
probability to
land to
These squares
(out with doubles)

Chance to Land at a
Square

Break Even Times

What we learned
 Staying in prison strategy alters changes to land
other squares
 Long prison stay good at the end game
 Break even time downward trend is good
 Breakeven times are long
 Slow start
 Note that one cannot build before owning all
squares with that color

Forbidden Island

Rules in brief
 Randomly generated boar,
 Treasure deck (28 cards)
 4 treasure cards are needed to
collect a certain (1/4) treasure
from specific tile
 Water rise! (3) cards increase
speed which the island is going
under water
 Flood cards
 Tells which tile will be flooded
or sink
 Players has three actions
 Support a tile
 Move
 Give a treasure card
 Capture a treasure
 Win by collecting all four treasures
and escape by helicopter
 Loose by
 Water level raises too high (with
Water Rise! Cards)
 Cannot collect the treasures
because of sunken tile
 Cannot escape because of
the exit tile is sunken

Game length (loose with
water level)
 3 water raises cards in treasure cards deck (28
cards)

Game length (loose with
water level)
Loose cond.:
N water rise!
Deck
exhausted N
times
Ends with
Nth water
rise! card
Novice 9 2 3
Normal 8 2 2
Elite 7 2 1
Legendary 6 1 3
Rough estimate about play time in terms of water rise!
cards

2 players, normal
Water
Level
Min turn Max turn Mean
turn
# act. /
support
Actions -
# actions
1 1 10 2.7 2 2
2 1 11 4.8 2 2
3 1 11 7.3 3 1
4 2 19 11.4 3 1
5 13 20 15.5 3 1
6 13 20 17.4 4 0
7 14 28 21.0 4 0
8 22 29 24.5 5 -1
9 22 29 26.4 5 -1

Prevent island sinking
 Water levels 1 & 2: possible to support squares most of the
time
 Water levels 3–5: possible to support nearby squares if actions
are focused to that
 Spending max 1 point to movement
 Water levels 6-7: Not possible to support all squares except
with luck
 No movement possible if supporting four squares
 Water level 8: island is sinking no matter what
 Note: Digging up a treasure requires an action and moving to
the correct square
 To keep the island in stable state would require more actions

Treasure cards
 4 same treasure card is needed to dig up a
treasure
 5 same treasure cards in deck
 Discarding correct cards is critical (max hand
size: 5 cards)
 Treasure card scuffle is needed if discarding 2 same
resources before the set can be completed

Summary

Balance
 Symmetry typically leads to balance
 Note: turn order / some typically start in board games
 Creates imbalance
 Catan solution to balance set-up
 turn order in setup: 1-2-3-4-4-3-2-1
 Symmetry can be also in form of rock-paper-scissors
 Balancing weapons & troops
 non-symmetrical things are harder to balance
 Difficulty in co-op games:
 resources needed to keep status quo or
 to progress vs resources available

Simulations
 Game systems with random component are
complex to predict
 Card-based are even history-dependent
 How many / what cards are played influence
probabilities
 Simulations can help to understand how a part of
the system behaves
 One does not need ready game for simulation
 But one needs to understand what to simulate
 Does not replace playtesting
 But simulation can show the features work in the long
run

Assignment

Assignment
 Use relevant presented approaches to analyze
your game design and
 Balance it / set difficulty
 Fine-tune play time
 Combine with play-testing
 Return
 Documentation of your process (steps, calculations)
 Around 1-2 pages

That’s all folks

Analysis for design

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