analysis for design

Analysis for DesignEvaluating 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 Systemskilld10 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

su

ccess

skill

1

2

3

4

5

6

7

8

9

100

10000

20000

30000

2 4 6 8

difffa

il

skill

1

2

3

4

5

6

7

8

9

10

0

2500

5000

7500

10000

2 4 6 8

diff

fum

ble

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


TroubleA 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


AssignmentIsland 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

Bo

ard

Petri Lankoski Södertörn University 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


Settlers of Catan

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 randomfrom collections import Counter

# board model (2 victory points) field1 = {

2: {'white': 0, 'blue':0, 'red': 0, 'orange': 0},3: {'white': 0, 'blue':0, 'red': 1, 'orange': 1},4: {'white': 1, 'blue':1, 'red': 0, 'orange': 0},5: {'white': 0, 'blue':2, 'red': 1, 'orange': 0},6: {'white': 1, 'blue':1, 'red': 1, 'orange': 1},8: {'white': 1, 'blue':1, 'red': 1, 'orange': 1},9: {'white': 1, 'blue':0, 'red': 0, 'orange': 1},10: {'white': 1, 'blue':0, 'red': 1, 'orange': 1},11: {'white': 0, 'blue':0, 'red': 1, 'orange': 1},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


Sta

rt p

osi

tio

n

co

mp

ariso

n

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

Reso

urc

es

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

Reso

urc

es

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

Re

sou

rces

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

Re

sou

rces

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)


Imp

ac

t o

f se

tup


Imp

ac

t o

f se

tup


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

Reso

urc

es

White, bad

3

4

5 5

0 0 0 0

1 1 1

2

3 3 3

4

1.80872.11382

2.424242.72642

0

2

4

6

0 1 2 3

Turn

Reso

urc

es

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

Re

sou

rces

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

Re

sou

rces

White

What this mean?


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

Fe

ed

ba

ck lo

op

& r

oll

“7” e

ffe

ct


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

Reso

urc

es

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

Reso

urc

es

Robber=No

6

11

12

14

1 1

2 22

3

4

54

6

8

10

3.27528

5.04646

6.527

7.68936

0

5

10

15

0 1 2 3

Turn

Re

sou

rces

Robber=To zero

6

11

12

14

1 1

2

3

2

3

4

54

6

8

10

3.29602

5.0909

6.50472

7.7635

0

5

10

15

0 1 2 3

Turn

Re

sou

rces

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

Data & Analytics