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骨灰级老玩家详细分析 关于力地府的理解【阴曹地府】
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　　导读：骨灰级地府老玩家详细分析，关于力地府的理解，以及对于力地府提出的N种修改方案。　　一、猜测策划想法：　　策划心里想的是注重游戏公平性原则。　　比如：DT四刀下去，2000一下，4刀8000有了。　　LG秒7下去，1000一个，7个单位7000血，比DT少了1000。　　但是DT的优点是暴发性强，伤害突出于1个点，LG是绵延不断，伤害输出突出一个面。　　我4刀下去，你LG挂了，于是无法进行输出，我休息的时候你拉起来秒我了，于是你LG有了所谓的输出。　　如果用大量数据推演就可以得出结论：各个门派的输出其实都是在“可预见范围内受控的”。　　好比是拿狙击***和加特林比较，没可比性，看你怎么用。我点杀不可能用龙卷，我面伤不会用横扫。　　梦幻前期的卡片，你得到一个技能，那么你必须有所付出。　　骷髅怪卡片，蝴蝶仙子卡片到后面的火车卡，混沌兽卡片等。　　咒师秒的多加血能力比不上莲台，莲台辅助强了，秒不过咒师也正常。(金刚和尚和杏林的也是一个德行)　　咒师开花了加血能力进一步打压，但是输出能力集合系数增长。(哪怕是现在，点选单位还是很凶)　　浴血和无双，同样1W血，扫下去，谁掉的血多?　　策划的所谓公平：即有得必有失　　地府的原始筋脉猜测：　　小死亡、九幽、符石这3个出来之前DF组队基本没戏，所以那时候很多DF都选择自己带队。于是造成现在很多高级别DF带队一个比一个狠...咱先天基础好扯远了。　　如果那时候力DF的不出现，那么目前DF的经脉应该就是和PT一个德行，拷贝不走样，一个主比武，一个主任务。　　但是，but，我们有力DF。所以这个神奇的经脉出现了。　　网易推出这个经脉之前，策划肯定做过测试，毋庸置疑，但是，但是，但是!　　测试的对象是什么?我一直在怀疑。　　思前想后，我猜测的是：策划自己在设计这个经脉的时候，做的游戏数据测试用的是DF的巅峰数据和其他物理门派比如ST的巅峰数据做比较。　　同样都是2000伤害，大家砍下去都是1000血，咦，不错不错，这个经脉就这样放出去了。　　必中的百爪，无伤害降幅的百爪。　　咦，不对啊，ST2回合鸟6，没套子3回合鸟6，DF2回合8个，3回合12了。不行不行，nonono，我得改　　怎么改?对了，有必中武器的，那就让力DF收必中去....阿嘞?这也不行啊，这样还是不划算，得~咱把伤害折损下~　　吾~哟西，不错不错，完事了。　　这里我怀疑策划做数据测试的时候，用的是DF晚上的数据和其他门派白天的数据做对比的。　　在输出方面折损的那么厉害的原因有2个：第一是任务砍不动的时候有阎罗令的存在，第二是无论任务还是打架，力DF能毒。自带鬼眼抗封+感知　　策划的想法是，增加力DF干架时候的机动性灵活性。　　而且最最最终于的经脉无忌，力DF是没有的。　　策划想的很简单：天生70%，鬼眼，鞋子，再给你无忌还了得!?这个物理输出门派没人能限制了。　　但是正是如此，DF门派的特色把力DF坑到西伯利亚去了......　　现在的策划，似乎一直在引导玩家，往哪边发展，看起来就像是个山西煤老板在教我们怎么样挖煤能够挖的又好又快....　　结果我们只是一次又一次的当了他的小白鼠。　　二、玩力DF必须要正视的规则　　1、DF的门派特色是 具备夜战能力，抗封70%　　后半条傻子都知道，满修情况下确实如此(运气差被秒封的没办法)　　前半条的意思是：DF的夜战能力是在白天的基础上，上升，记着，我们的战斗力在晚上是上升的。　　不是在白天战斗力下降了。　　这一条是铁的，不是文字游戏，也不是帮策划那个坑货找借口，力DF成也特色败也特色。搞不清楚的，就不要去想怎么去强大力DF。　　从这里入手去改的，整个DF都要被掀的底朝天，门派特色策划绝对不会退步的。　　2、召唤兽的夜战和卡片附带夜战技能　　这个夜战的意思是：夜间物理伤害不会减少(无此技能的夜间伤害将减少20%)　　3、综上所述，大家伤害都是1000的，到了晚上，其他门派都是800了，力DF是1200了，人家变身夜战卡，依旧1000伤害。就这么简单。　　4、力DF因为和血DF一样享受了如此奇葩的夜战特色，于是我们的属性版面的伤害数字白天晚上也会有浮动。为什么其他门派的属性板面不会有浮动?我想着应该归功于夜战卡片。这个卡片导致策划不想去改。　　三、力DF手术方案　　方案1、人家都有“夜战”卡片，那么给力DF“白战”卡片　　这一条看起来如此美好，但是实际上可行性比较低在10%左右　　分析：出一张“白战”卡片的代价是：新的召唤兽要出来，并且这个召唤兽还得很奇葩的携带一个技能，白战这意味着这个召唤兽自身战斗力也具备和DF一样的特色...或者说白天这奇葩召唤兽的战斗力减弱，有了白战，才会保持晚上的战斗力。　　而且白战的定义是什么?保持晚上的战斗力?那就恭喜了，这张卡片绝对是DF双系都欢迎的卡片，15个门派单独为一个门派设计一张卡片，这***，策划一般不会去做。　　或者策划会考虑拿现成的BB去更改做附加技能，比如蛟龙卡片(感知+永恒)　　但是这次新资料片有新BB放出，新地图，新玩法....我还是有所期待。　　不过话说，新BB出来变身要求应该在6级，低级别还是比较苦，高级别任务的问题解决了，但是PK的问题还是有很大的。我算你159的2800伤害好了，伤害折损后你还砍的动谁?　　69,89可以靠套装解问题，109后套装成本加大，但是109和129的力DF灵活性还是很有问题的。无限六道?　　DF既然定位诅咒系了，那么血耐敏DF的诅咒特色发挥出来的同时，力DF的诅咒系输出能力能否增加?　　方案2、新三级药出来了。　　所以可以增加一种“秘药”或者是在烹饪那边改一下，增加一种类似于“福灵沙”“同心肉脯”一样的烹饪或者3级药，服用后在一定时间内保持夜间的战斗力振幅。　　这个方案的可行性个人感觉还是比较高的，15%左右　　不过就怕那些神奇的喷们，碾转于各大论坛，毫不知疲倦如抽水机般日日夜夜哗啦呼啦的口水像不要钱一样喷薄而出。　　分析：既然针对DF有这种方案进行战斗力振幅，那么我们DT来个道具让我保持战斗力不下降就好了，我比武可以变猫灵抢速了，可以变踏云兽撑血了。　　方案3、新增剧情。　　召唤兽都有剧情了，你策划15个门派30条经脉的剧情你做起来也不会多累的。让文案多加几天班，写个30条经脉剧情方案好了，想点那个脉的就做哪个脉的经脉，做完剧情才能点脉。　　这里适当可以增加一些先决条件，因为个别门派目前是2个脉一起点的。30%　　分析：魔兽争霸中的恶魔猎手弄瞎了自己的双眼，从而获得了可以迅速发现恶魔的能力。　　梦幻西游中的力地府，弄瞎了自己的双眼，从而获得了永久性不惧怕白天的能力。　　但是在这里我要提出的是：如果方案可行，力DF的索命套最好无法使用，并且阎罗令平秒能力减半或者打折扣。换来的代价是百爪，六道无量的伤害高一点。　　方案4、力DF奇经八脉大改。　　地赐3 使用法术减少2,4,6点蓝　　赦震3 反震气血损失减少15,30,45　　狂力3 人物力量增加4,8,12　　地赐和赦震其实可以2择1考虑改成“厉鬼：白天伤害增加5%，阎罗令，判官令技能伤害减少25%”　　局部放大DF门派特色，但是任务烧双能力减弱，逼迫DF2择一经脉，这样的后果是部分低级别力DF会选择血DF经脉，低级别装备一旦占优势战术很灵活多变，辅助输出转变就在一瞬间。　　这样的好处是力DF放弃了饭碗技能，阎罗令离你而去，彻彻底底的让你去力了。换来经脉的后期爆发。　　凝目4 增加8,16,24,32点命中　　毒云4 攻击尸腐毒目标增加20,30,40,50点物理伤害　　幽体4 遭受法术攻击时有40%的几率降低20,40,60,80点法术伤害　　既然属诅咒系输出，毒云这个技能其实已经可以形成双系DF战场共存的情况了。　　幽体感觉可改为：物理攻击时候有10%20%30%40%的几率驱散对方一个良性状态or增加一个尸腐毒效果(前者增加战场随机性，后者使得109往后队伍中双系DF存在有所疑问，血条吃紧的时候辅助加血能力减弱。)　　百炼3 武器耐久消耗降低20%，35%，50%……请各位alt+v 输入“大唐门派”特色你就会觉得，前期其实策划还是有所照顾力DF的，一个筋脉搭上一个门派特色　　六道无量3 攻击时附带六道轮回技能的20%40%60%伤害　　淡定4 不解释　　夜行4 增加人物12,18,24,32点速度(这一点感觉策划先期考虑的时候就在想让力DF作为诅咒型输出配合DT的，同样情况下，力DF速度就比DT快了。)　　随着新门派的出现，策划自身也进入一个怪圈——无亮点技能出现(饭碗技能太过单调)　　MW如果说是法爆，那么LG可以说是魔心，接着新来的SML那就是法连特色　　FC的封降防御，NE老牌的一速封，WZ的三封 都是封系特色，新来的无底洞则是融合了回血、状态、封的特色，速度基本卡在血耐敏DF一个档次　　而凌波城这个物理系则是苦逼的有了一个新的概念“战意值”技能限制较高，而且更加容易被控制。　　那么这种时候，力DF绝对是一种诅咒系输出的不二人选，六道无量肯定要改啊。　　不为别的，59,69,89靠装备补，109后同样都是玩家，我们不可能继续依靠套装的“血轮眼”功能移植其他门派的技能来混饭吃。这是底线。　　“六道无量：有分地狱道、饿鬼道、畜生道、修罗道、人间道、天界道，每级增加2道，1级100%攻击1下，2级为50%出现第二下，3级为25%出现第三3下　　每次攻击进行判定，有可能是队友，有可能是对手(第一下必为对手)　　攻击对手了，那么则为输出，输出的时候每砍对手一下就判定一下，几率就像必杀善恶一样　　一级：地狱道自身伤害的200%攻击，饿鬼道120%伤害攻击附带吸血能力(二择一随机攻击)　　二级：畜生道对召唤兽造成150%伤害，修罗道80%伤害攻击2个目标(二择一随机攻击)　　三级：人间道对队友进行恶性buff驱散，天界道为队友增加一个良性buff(二择一随机出现)　　也就是说对于109DF来讲，有个强力的输出技能当饭碗了。”　　鬼火2 每次使用特技后获得1，2点愤怒　　阴凝4 每次使用非物理攻击法术后获得5，10,15,20点伤害提升　　伤劫3 使用六道无量的时候增加武器总伤的6%12%18%的伤害攻击　　4层的经脉其实过渡大于实用，个人认为可以把6层的隐咒拿来做替换，　　隐咒：队友在你附加的隐身状态下可以使用法术，但是使用完后隐身解除并且所受伤害增加25%　　隐咒换伤劫，或者鬼火，队友(包括自己)在你附加的隐身状态下可以使用法术，但是使用完后隐身解除并且所受伤害增加50%，25%(3层或者是50%35%20%)　　因为是所受伤害增加50%比如我砍/秒你1000，变成1500，这个可以接受。　　这并不是WZ的天地同寿，防御减少50%，那个玩大了，我2000攻击砍不动你2000防御的我难道还砍不动你个1000防御的?　　增加了109往后的力DF的灵活性，配合防点杀能力。　　阴魂3 普通物理攻击时候忽视25,50,75防御　　骨爪4 百抓狂杀伤害折损率16%15%14%13%　　隐遁4 己方所以队员在修罗隐身状态下所受所有伤害减少8,16,24,32　　百抓狂杀4 1,2,3,4个目标，非必中，伤害有折损率　　阴魂的出现我很意外，策划这里想说明什么?让159后的DF继续穿套子平砍去?至少要把普通2个字去掉吧?　　骨爪感觉可以改成，百抓狂杀攻击目标时候有20%30%40%50%给目标造成恶性buff(红毒，碎甲刃，八凶阵法，些许伤势或锢魂术)　　隐遁4级，配合之前的隐咒修改。隐咒多几个乾元丹那就在隐遁这里扣掉几个。　　百爪 1234个目标，命中率65%75%85%95%，伤害折损率16%15%14%13%　　用额外的附加状态来弥补因为百爪伤害不足所造成的输出不够问题。　　气感3 使用百爪狂杀时，临时提升武器10%20%30%的命中。　　隐咒1 队友在你附加的隐身状态下可以使用法术，但是使用完后隐身解除并且所受伤害增加25%　　汲魂4 每击杀一个召唤兽or人物时候，增加4,8,12,16伤害以及1%锢魂术命中率　　击破3 攻击目标在防御状态的增加4,8,12%的伤害效果　　气感个人的理解是砍人的时候，我的武器命中是1000，那么就增加10%20%30%的命中就是100,200,300也就是说最高100点的伤害。如果我理解没错，这个其实就是在弥补百爪的伤害问题的，策划这里应该在替高级力DF擦屁股。　　渡劫后很多力DF都血耐敏了，但是还有很多力DF在坚持。　　隐咒不多说，这个JN我认为铁定要放前面去，但绝对不是放在前三层，防止个别DF走双修，那乐子就大了。　　莲花心音都丢3层了，隐咒不丢前面去太浪费了。　　汲魂：这个怎么说呢，那个1%的锢魂术是什么意思?让力DF魔武双修?法修也去点?　　这里我感觉，如果真的牺牲老牌烧双饭碗技能阎罗令，那么小死亡其实也没必要继续补了。经脉里可以用“强迫性”过渡技能来压制力DF小死亡。从而增加双系DF一个队伍配合的可行性。主观上消除队伍配置的问题，促进力DF融合进入比武队伍。(都能比武了任务队肯定没问题啊)　　无常步下面增加一个力DF御用技能，这样在飞升的时候可以6JN130让双系DF有个选择性　　击破：有这个闲工夫砍人家防御的，我宁愿你再给我来个和壁垒差不多的技能。　　能点到击破的DF，百爪和六道都会了，谁吃饱了撑着会去砍防御状态的玩意。ST这年头第一偷不到。第二偷到怎么样?力DF不大改，伤害就上不去，砍人家1000，多了120血....　　这样其实不如改成攻击防御状态的目标时候，有30,60,90%几率使用一次幽冥鬼眼分别增加2,4,6个(类似于FC的凝神和ST的极度疯狂)　　夜之王者2 百爪狂杀附加六道轮回技能等级*30%60%的伤害以及必杀几率5%10%　　后记：　　个人认为，力DF在任务输出方面的“不给力”是因为还保存了血DF时期的阎罗令，索命套等。如果能够以降低阎罗令等一系列任务烧双饭碗技能的前提下(即将阎罗令等变成花瓶技能为代价)提高DF在白天的伤害乃至像夜晚看齐。这里解决了任务时候出现的双系DF的尴尬。血敏DF照样可以辅助力DF，而经脉中可以削弱九幽回血代价来增加力DF伤害，或者让九幽也可以成为主动法宝。这里也不用担心哪个想不开血DF下了九幽玩力的。而摄魂作为共存法宝，让力DF的诅咒能力也提升些许(改成队友敌方都能丢?敌方降防降攻，友方提升防御防止点杀?)　　其次，力DF在PK时候的作用，低级别装备补，WJB武器怎么砍都疼。但是一旦到109往后，装备优势变成劣势，势必需要新的方向来弥补。那就是新的饭碗技能。　　而有之前白天伤害的提升做保障，那么新技能的出现也是无可厚非的，诅咒系突出特点灵活多变，用来恶心人。　　六道无量的技能突出了DF清场点杀BB能力及变相辅助抗点杀能力，隐咒用来增加战场灵活性多变性。
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网易通行证The St. Petersburg Paradox (Stanford Encyclopedia of Philosophy)
The St. Petersburg Paradox
The St. Petersburg game is played by flipping a fair coin until it
comes up tails, and the total number of flips, n, determines
the prize, which equals $2n. Thus if the coin comes
up tails the first time, the prize is $21 = $2, and the game
ends. If the coin comes up heads the first time, it is flipped again.
If it comes up tails the second time, the prize is $22 = $4,
and the game ends. If it comes up heads the second time, it is flipped
again. And so on. There are an infinite number of possible
&consequences& (runs of heads followed by one tail)
possible. The probability of a consequence of n flips
(P(n)) is 1 divided by 2n, and the
&expected payoff& of each consequence is the prize times
its probability. The following table lists these figures for the
consequences where n = 1 & 10:
Expected payoff
(Only the start of the full table, which is infinite.) The
&expected value& of the game is the sum of the expected
payoffs of all the consequences. Since the expected payoff of each
possible consequence is $1, and there are an infinite number of them,
this sum is an infinite number of dollars. A rational gambler (who
values only money and whose desire for any extra dollar does not depend
on the size of her fortune) would enter a game iff the price of entry
was less than the expected value. In the St. Petersburg game, any
finite price of entry is smaller than the expected monetary value of
the game. Thus, the rational gambler would play no matter how large the
finite entry price was. But it seems obvious that some prices are too
high for a rational agent to pay to play. Many commentators agree with
Hacking's () estimation that &few of us would pay
even $25 to enter such a game.& If this is correct&and if
most of us are rational&then something has gone wrong with this
way of thinking about the game. This problem was discovered by the
Swiss eighteenth-century mathematician Nicolaus Bernoulli and was
published by his brother Daniel in the St. Petersburg Academy
Proceedings (1738; English trans. 1954); thus it's called the St.
Petersburg Paradox.
Bernoulli's reaction to this problem was that one should distinguish
the utility&desirability or satisfaction production&of a payoff
from its dollar amount. His approach in a moment: but first consider
this way (not his) of trying to solve this problem. Suppose that one
reaches a saturation point for utility: that at some point, a larger
dose of the good in question would not be enjoyed even a little bit
more. Consider someone who loves chocolate ice-cream, and every
tablespoon of it he eats gives him an equal pleasure&equivalent to
one utile. But when he has eaten a pint of it (32 tablespoons), he
suddenly can't enjoy even a tiny bit more. Thus giving him 16
tablespoons of chocolate ice-cream would provide him with 16 utiles,
and 32 tablespoons with 32 utiles, but if he was given any larger
quantity, he'd enjoy only 32 tablespoons of it, and get 32 utiles. Now
imagine a St. Petersburg game in which the prizes are as above, except
in tablespoons of chocolate ice cream. Here is the start of the table
of utiles it provides:
Utiles of Prize
Expected utility
The sum of the last column now is not infinite: it asymptotically
approaches 6. In the long run, a player of this game could expect an
average payoff of six utiles. A rational ice-cream eater would pay
anything up to the cost of 6 tablespoons of chocolate ice-cream.
I've chosen ice-cream for that example because that sort of
utility-maxing might be considered less implausible than when other
goods are involved. Compare the original case, where payment is in
dollars. Replacement ($X for X tablespoons) shows a maximum utility reached by
any dollar prize over $32, and a fair entry cost of $6. But putting the
maximizing point there would mean that $10,000 was worth no more&provided no more utiles&than $32. Nobody feels that way. So should
we raise the point at which an additional dollar has no additional
value? Setting that point at (say) $17 million is better. It's such a
large prize that you might feel that anything larger wouldn't be
better. Setting the maximum point there, by the way, means that the
maximum rational entry price for this sort of game would be a little
above $25, which is Hacking's intuition about the reasonable maximum
price to pay. So is that the point where utility maximizes out, for
money? Agreed, that's a lot of money, and it would take some
imagination for some of us to figure out
has seemed to most people thinking about this sort of thing that there
isn't any point at which an additional dollar means literally nothing&confers no utility at all. It seems that the-more-the-better desires
are possible&even common.
But that does not mean that the expected value of St. Petersburg
must be infinite. There are ways to approach this such that every
additional dollar confers additional utility, but the sum of the rows
is not infinite. The first proposal of this sort was due to Bernoulli
himself. The same paper in which he proposed this problem contains the
first published exposition of the &Principle of Decreasing Marginal
Utility,& which he developed to deal with St. Petersburg. This
principle, later widely accepted in the theory of economic behavior,
states that marginal utility (the utility obtained from consuming an
extra increment of the good) decreases as the quantity consumed
in other words, that each additional good consumed is less
satisfying than the previous one. He went on to suggest that a
realistic measure of the utility of money might be given by the
logarithm of the amount. Here is the beginning of the table for this
gamble if utiles = log($):
Expected Utility
The sum of expected utilities is not infinite: it reaches a limit of
about 0.602 utiles (worth $4.00). The rational gambler, then, would pay
any sum less than $4.00 to play. But there is no maximum utility for
any outcome: any of the possible outcomes on the list confers more
utility than the one above it.
Some have found this response to the paradox unsatisfactory, because
Bernoulli's association of utility with the logarithm of monetary
amount seems way off. On his scale, the utility gained by doubling any
thus the difference in utility between $2
and $4 is the same as the difference between $512 and $1024. However
there are other ways of discounting utility as the total goes up which
may seem more intuitively plausible (see for example Hardin 1982;
Gustason 1994; Jeffrey 1983). The main point here is that if
the sum of the utilities in the right-hand column approaches a limit,
then the St. Petersburg problem is solved. The rational amount to pay
is anything less than this limit. In his classical treatment of the
problem, Menger (]) argues that the assumption that there is an
upper limit to utility of something is the only way that the paradox
can be resolved.
However, decreasing marginal utility, approaching a limit, may not
solve the problem. Let us agree that money has a decreasing marginal
utility, and accept (for the purposes of argument) that a reasonable
calculation of the utility of any dollar amount takes the logarithm of
the amount in dollars. The St. Petersburg game as proposed, then,
presents no paradox, but it is easy to construct another St. Petersburg
game which is paradoxical, merely by altering the dollar prizes.
Suppose, for example, that instead of paying $2n
for a run of n, the prize were $10 to the power
2n. Here is the beginning of the table for this
Utiles of Prize
Expected utility
The expected value of this game&the sum of the infinite series
of numbers in the last column&is infinite, not just very
large. That means no amount of money is too large to pay for one game.
The problem returns.
Bernoulli's log function is not the only way to relate dollar values
to utility, but there is a general objection to all of them. Imagine a
generalized paradoxical St. Petersburg game (suggested by Paul Weirich
(), following Menger (])) which offers prizes in
utiles instead, at the rate of 2n utiles
for a run of n (however that number of utiles is to be
translated into dollars or other goods). This game would have infinite
expected value, and the rational gambler should pay any amount, however
large, to play.
But should we say that there's a limit to how many utiles
anyone can absorb? It seems not. Even if the utility per unit of every
particular good diminishes with quantity, this does not imply that the
capacity to appreciate additional utiles diminishes somehow. When
you're sick of ice-cream, you still can enjoy Mozart. The mega-rich
seem always to be able to find something new that they're interested
in. There is no law of the diminishing utility of utiles.
For simplicity, we shall ignore the generalized version of the game,
and continue to discuss it in terms of the original dollar prizes,
recognizing, however, that the diminishing marginal utility of dollars
may make some revision of the prizes necessary to produce the
paradoxical result.
Consider the following argument. The St. Petersburg game offers the
possibility of huge prizes. A run of forty would, for example, pay a
whopping $1.1 trillion. Of course, this prize happens rarely: only once
in about 1.1 trillion times. Half the time, the game pays only $2, and
you're 75% likely to wind up with a payment of $4 or less. Your chances
of getting more than $25 (the entry price which Hacking suggests is a
reasonable maximum) are less than one in 25. Very low payments are very
probable, and very high ones very rare. It's a foolish risk to invest
more than $25 to play.
This sort of reasoning is appealing, and may very well account for
intuitions that agree with Hacking's. Many of us are risk-averse, and
unwilling to gamble for a very small chance of a very large prize.
There are a couple of ways of factoring in this risk-aversion. One
way is building it into the utility function, so that the utility of
(for example) a sure-thing $10 is higher than the utility of a 50%
chance of $20. Another way counts them as the same, but adds in the
negative utility of the anxiety of prior uncertainty. Weirich (1984),
claiming that considerations of risk-aversion solve the St. Petersburg
paradox, offers a complicated way (which we need not go into here) of
including a risk-aversion factor in calculations of expected utility,
with the result that there is a finite upper limit to the rational
entrance fee for the game.
But there are objections to this approach. For one thing,
risk-aversion is not a generally applicable consideration in making
rational decisions, because some people are not risk averse. In fact,
some people may enjoy risk. What should we make, for example, of those
people who routinely play state lotteries, or who gamble at pure games
of chance in casinos? (In these games, the entry fee is greater than
the expected utility.) It's possible to dismiss such behavior as merely
irrational, but sometimes these players offer the explanation that they
enjoy the excitement of risk. Differences among people in their
feelings about risk (and, for that matter, in how much decrease in the
marginal utility of money they experience) might be accommodated by
allowing utility functions individually tailored to particular
personalities. But in any case, it's not at all clear that
risk-aversion can explain why the St. Petersburg game would be widely
intuited to have a fairly small maximum rational entry fee, while so
many people are willing to risk very large sums of money for highly
unlikely huge payoffs in other games.
But even if we assume, for the purposes of argument, that
risk-aversion is responsible for the intuition that the appropriate
entrance-fee for the St. Petersburg game is finite and small, this will
not make the paradox go away, for we can again adjust the prizes to
take account of this risk-aversion. It seems likely that someone who
didn't like to gamble would play if the prize were increased enough.
The fact that many more people enter lotteries when unusually big
prizes are announced, keeping risk more or less constant, is evidence
But perhaps these considerations do not apply to all aspects of the
St. P if it has infinite utility, then even a very high
entry price is justified, and the risk of losing that amount is very
high. The most compelling examples of the rational unacceptability of
risk no matter how high the prize, are the ones in which the entry
price is high and the prize improbable. Imagine, for example, that you
are risk averse, and are offered a gamble in which the entry price is
your life-savings of (say) $100,000, and the chances of the prize are
one-in-a-million. It seems rational to refuse, no matter how huge the
prize. Perhaps that is what explains the unwillingness to make a big
investment in St. Petersburg. But note however that this may not be a
sufficient response to the paradox. This sort of risk-aversion would
also provide a psychological explanation of why (some) people are
unwilling to gamble large sums when the finite expected
utility is greater than the initial payment. So, for example, many
people would be unwilling to risk $1000 for a one-in-a-hundred chance
at winning $200,000 (expected value $2000). If risk is not a disutility
that can be compensated for by prize increase, then maybe their
behavior runs counter to the expected-value theor
and if they're rational, then maybe this shows the theory is wrong. The
paradox raised by St. Petersburg, however, is not thereby fully dealt
with. It is not merely a case&others of which are well-known&in
which apparently rational behavior disobeys the advice to maximize
expected value. What's paradoxical about the St. Petersburg
paradox is that its expected value is infinite.
Gustason suggests that, in order to avoid the St. Petersburg
problem, one must either restrict legitimate games to (a) those that
have consequences with an upper bound on values, (the possibility we've
been looking at), or to (b) those in which each act has only finitely
many consequences. Let us consider the imposition of restriction
Imagine a life insurance policy bought for someone at birth, which
pays to that person's estate, at eventual death, $100 for each birthday
the person has passed, without limit. What price should an insurance
company charge for this policy? Standard empirically-based mortality
charts give the chances of living another year at various ages. Of
course, they don't give the chances of surviving another year at age
140, because there's no empirical evidenc but a
reasonable extension of the mortality curve indefinitely beyond what's
provided by available empirical evid this curve
asymptotically approaches zero. On this basis, ordinary mathematical
techniques can give the expected value of the policy. But note that it
promises to pay off without limit. If we think that, for each age,
there is a (large or small) probability of living another year, then
there are an indefinitely large number of consequences to be considered
when doing this calculation, but mathematics can calculate the limit of
and (ignoring other factors) an insurance company
will make a profit, in the long run, buy charging above this amount.
There's no problem in calculating its expected value. This is not a
&game,& of course, but this casts doubt on the necessity of restricting
the consideration of probabilities and payoffs to cases with finite
consequences.
Because it's often insisted that St. Petersburg, as described, is
not suited for considerations given for ordinary games, thinking
sensibly about it requires modifications in the way it would work&modifications we would expect if this game were really offered in
casinos. One way to do this is to assume that the way the game would
not run exactly as described. For example, the casino might terminate
play after some number of consecutive heads, call it N, and pay off for
a run of N, even though it had not been ended by throwing tails. Thus
there would be only a finite number of possible consequences. If N were
set at 25, then the game would have an expected value of $25, and that
would be the maximum entry price which a rational agent would pay to
play (as in Hacking's intuition). Do we, perhaps unconsciously, assume
that casinos would truncate any run of 25 heads? Why 25?
Many authors have pointed out that, practically speaking, there must
be some point at which a run of heads would be truncated without a
final tail. For one thing, the patience of the participants of the game
would have to end somewhere. If you think that this sets too narrow a
limit N, consider the considerably higher limit set by the life-spans
of the participants, or the surviv or the limit
imposed by the future time when the sun explodes, vaporizing the earth.
Any of these limits produces a finite expected value for the game, but
sets an N which is considerably higher than 25; what, then, explains
Hacking's $25 intuition?
A more realistic limit for N might be set by the finitude of the
bankroll necessary to fund the game. Any casino that offers the game
should (it seems) be prepared to truncate any run that, were it to
continue, would cost them more than the total funds they have available
for prizes. A run of 25 would require a prize of a mere $33,554,432,
possibly within the reach of a larger casino. A run of 40 would require
a prize of about 1.1 trillion dollars. So any casino offering St.
Petersburg must truncate very long runs. If a state backing a casino
were crazy enough to print up enough money to pay off a colossal prize,
economic havoc would result, including massive inflation severely
reducing the utility of the prize. Might these practical difficulties
may be circumvented by a casino's following the suggestion made by
Michael Clark (2002) that an enormous win would be offered merely as
credit to the winner? Would anyone believe that promise?
It's of course true that any real game would impose some upper limit
N, producing a finite number of possible cons but
this does not solve the St. Petersburg puzzle because these finite
games are not St. Petersburg. Our question was about the St. Petersburg
game, not about its N-limited relative.
Russell's Barber Paradox is resolved by showing that there is
might realistic considerations resolve the St.
Petersburg paradox by showing there is no such game? Jeffrey (1983:
154) says that &anyone who offers to let the agent play the St.
Petersburg game is a liar, for he is pretending to have an indefinitely
large bank.& He modifies this by imagining the indefinitely large
bank of a government able to print up money at will, but points out
that inflation would then make the expected utility finite. If someone
without this government backing appears to be offering this game, then
Jeffrey claims this offer must be &illusory&: he's
not really offering St. Petersburg, but rather a Petersburgesque game
in which there will be an upper bound on winning.
Now, it's true that anyone who makes this offer should realise that
it includes some highly unlikely consequences he'd
but this is not lying. Someone who didn't realize that
his finite bank couldn't cover possible consequences might
sincerely offer the game, as might someone who realized that, but took
the risk of owing what he couldn't pay. The advice,
&Don't make bets you can't cover&&that
is, never bet without the ability to pay for the worst
outcome&has a point because sometimes people do make such bets.
The St. Petersburg bet can't be covered, but it can
seriously be offered. When someone offers a bet but doesn't pay
off because he can't, we don't conclude that the offer was
&illusory.& We persist in the belief that the offer was
Maybe Jeffrey's point is really to explain why people
(supposedly) wouldn't pay more than a modest amount ($25?) to
play: because they wouldn't believe they are being offered a
genuine St. Petersburg game. That, however, goes no way toward
&Resolving the Paradox& (Jeffrey's title for this
part of his book, p. 154). The paradox is that the real St.
Petersburg game has infinite desirability and is a bet that can't
be covered by the person offering. The fact that we wouldn't
trust anyone who appears to offer it doesn't
&resolve& anything.
Expected value is in effect average payback in the long run.
Consider the graph of average wins in a series of the simple game in
which you get $12 when a fair die comes up 6, nothing otherwise. After
one game is played, the average win is either $12 or $0. As games
continue, the line representing the average so far will go up and down,
but as more and more games are played, the line gets closer and closer
to $2. If the cost of playing each game is $1, and one is playing very
many games, one can count on coming out increasingly close to a net
gain (payout minus cost) of $1 per game. A very odd casino offering
this game is likely to lose on average very close to $1 per game, as
many customers play one or more games each. This average over the long
run is what we're supposed to consider when calculating the rationality
of p why the average result is relevant to the
rationality of playing a single game is an interesting philosophical
question about any sort of game, and need not concern us here. But even
if every customer plays only one game, we can see that in the long run
the casino will lose money, and the customers (as a group) will
The line graphing average wins for a series of these dice games can
be expected to swing up and down with decreasing amplitude, narrowing
in closer and closer to $2. But the line for a series of St. Petersburg
games will behave quite differently. It will characteristically start
off quite low&after all, in three quarters of the games, the
payoff will be either $2 or $4. But after a while there will be a
sudden spectacular jump upward, as an improbable but huge payoff
occurs. Following this, the line will gradually sink, but then will
jump upward for another huge payoff. These big jumps will occur
increasingly rarely, but big jumps will bring the general tendency of
the line increasingly higher. A typical graph can be seen below (Hugh 2005).
Unlike the dice game's graph, the one for the St. Petersburg game
the longer a series of plays, the larger the
average win (roughly speaking). The more games played in a casino, the
larger the casino's average payout per game will get. The casino will
lose in the long run, and the customers as a group will gain, no matter
what price is charged per game.
All this is just a recasting of what we've already talked about,
from a slightly different angle. But this way of looking at things
suggests some conclusions that follow, and some that don't.
First: note that if you're going to play St. Petersburg with a
substantial fee per game, you're likely to have to play a very very
long time before you come out with a positive net payoff. Ordinary
practical considerations thus apply, as we've seen, but in theory,
you'd make a net gain if you continued long enough.
In this respect, playing St. Petersburg is rather like the
Martingale Strategy for games of chance. Here's a version of
Martingale. In advance, set a target for net winnings (winnings minus
losses). Bet an amount on the first game such that if you win, you'll
meet your target. If you win, go home. If you lose, bet an amount on
the next game such that if you win, you'll get an amount equal to your
loss on the first game plus your target. If you win, go home. If you
lose, bet an amount on the next game such that if you win, you'll get
an amount equal to all your losses so far plus your target. Continue
this till you win. One win will give you a net gain of your target.
This Martingale strategy would always work, if pursued long enough.
But of course there are practical difficulties. You may have to play a
very long time before winning once. But worse: in a series of losses,
the amount needed for the next bet rises. How much bankroll would you
need to guarantee success with the Martingale? Whatever bankroll you
take into the casino, there's a chance that this will not be enough for
a successful Martingale. Similarly, whatever the cash reserves of the
casino, there's a chance this will not be enough to pay off a big St.
Petersburg winner.
So we should be clear about both St. Petersburg and Martingale. With
a finite very large bankroll, a Martingale player might be likely to
win, but there's a chance that he'll run out of money to invest before
that. In practical terms, the Martingale player could be
certain of winning only if he could be certain that his
bankroll would survive any number of consecutive losses. And he
can't be certain of this. Similarly, with a finite very large
reserve, a casino would be likely to make money offering a
high-entry-payment St. Petersburg, but there's a chance, increasing
with more plays, that it will be caught with insufficient funds.
The St. Petersburg game is sometimes dismissed because it is has
infinite expected value, which is, it's argued, not merely practically
impossible, but theoretically objectionable&beyond the reach even
of thought-experiment. But is it?
Imagine you were offered the following deal. For a price to be
negotiated, you will be given permanent possession of a cash machine
with the following unusual property: every time you punch in a dollar
amount, that amount is extruded. This is not a withdrawal from your
neither will you later be billed for it. You can do this as
often as you care to. Now, how much would you offer to pay for this
machine? Do you find it impossible to perform this thought-experiment,
or to come up with an answer? Perhaps your answer is: any price at all.
Provided that you can defer payment of the initial price for a suitable
time after receiving the machine, you can collect whatever you need to
pay for it from the machine itself.
Of course, there are practical considerations: how long would it
take you to collect its enormous purchase price from the machine? Would
you (or the machine) be worn out or dead before you are finished? Any
bank would be crazy to offer to sell you an infinite cash machine (and
unfortunately I seem to have lost the address of the crazy bank which
has made this offer). But so what? The point is that there appears to
be nothing unthinkable about this thought experiment. It imagines an
action (buying the machine) with no upper limit on expected value.
It seems unlikely that your intuitions tell you to offer (say) $25
at most for this machine. But the only difference between this machine
and a single-play St. Petersburg game is that this machine guarantees
an indefinitely large number of payouts, while the game offers a
one-time lottery from among an indefinitely large number of possible
payouts, each with a certain probability. The only difference between
them is the probability factor: the same difference that exists between
a game which gives you a guaranteed prize of $5, and one which gives
you half a chance of $10, and half a chance of $0. The expected value
of both the St. Petersburg game and the infinite cash machine are both
indefinitely large. You should offer any price at all for either.
We'd suspect, in real life, that the offer of either was
illusory. But nevertheless it seems that the notion of infinite
expected value is at least thinkable.
Of course, that notion does create havoc. Suppose you assign
infinite value to going to heaven. God tells you that you'd
increase your chances of going to heaven, currently zero, by 1% for
every good deed you do from now on, making heaven certain after the
hundredth. The expected value of doing only one good deed is .01 &
&. But this equals the expected value of doing 100 good deeds:
1.00 & &; so you should help that old person across the street,
and then lapse back into your old complete selfishness. But that
conclusion is crazy. (Jeffrey () gives what's
basically this example under the title &De contemptu
And unbounded parameters can play havoc with considerations of
rational choice. Here's a practical example of that from real life.
Since home computers were introduced, they have steadily been getting
better and cheaper. Should you buy a new one now? No, it's better to
make do with the one you have now, and wait a few months till new
better and cheaper ones come out. But, on the assumption that this will
always be the case, it's always so you'll never
buy a new one no matter how good and cheap they get. Where's the
mistake in this reasoning? We can avoid this paradox by introducing
some &realistic& considerations, not mentioned in the story given, as
we can for St. Petersburg. But if we don't change the story or
introduce additional information, crazy things follow from the computer
story and St. Petersburg. That's why they're both
paradoxes.
The St. Petersburg game is one of a large number of examples which
have been brought against standard (unrestricted) Bayesian decision
theory. Each example is supposed to be a counter-example to the theory
because of one or both of these features:
the theory, in the
application proposed by the example, yields a choice people really do
not, thus it is descriptively inadequate.
theory, in the application proposed by the example, yields a choice
people really ought not to make, or which a fully, ideally rational
person, thus it is normatively inadequate.
If you see the standard theory as normative, you can ignore
objections of the first type. People are not always rational, and some
people are rarely rational, and an adequate descriptive theory must
take into account the various irrational ways people really do make
decisions. It's no surprise that the classical rather a-prioristic
theory fails to be descriptively adequate, and to criticize it on these
grounds rather misses its normative point.
The objections from standpoint (2) need to be
and we have been treating the responses to St. Petersburg as cases of
this sort. Various sorts of &realistic& considerations have
been adduced to show that the result the theory draws in the St.
Petersburg game about what a rational agent should do is incorrect.
It's concluded that the unrestricted theory must be wrong, and that it
must be restricted to prevent the paradoxical St. Petersburg
When considering the plausibility of restricting expected value
calculations in various ways that would take care of the paradox. Amos
Nathan () remarks,
it ought, however, to be
remembered that important and less frivolous application of
[unrestricted-value] games have nothing to do with gambling and lie in
the physical world where practical limitations may assume quite a
different dimension.
Nathan doesn't mention any physical
applications of analogous infinite value calculations. But it's
nevertheless plausible to think that imposing restrictions on theory to
rule out St. Petersburg bath water would throw out some babies as
Any theoretical model is an idealization, leaving aside certain
practicalities. &From the mathematical and logical point of
view,& observes Resnik (), &the St. Petersburg
paradox is impeccable.& But this is the point of view to be taken
when evaluating a theory per se (though not the only point of view ever
to be taken). By analogy, the aesthetic evaluation of a movie does not
take into account the facts that the only local showing of the movie is
far away, and that finding a baby sitter at this point would be very
difficult. If aesthetic theory tells you that the movie is wonderful,
but other considerations show you that you shouldn't go, this isn't a
defect in aesthetic theory. Similarly, the mathematical/logical theory
for explaining ordinary casino games is not defective because it
ignores practicalities encountered in making a real gambling decision.
For example, in deciding whether to raise, see, fold, or cash in and go
home, in a particular poker game, you must consider the facts that it's
5 A.M. and you are cross-eyed fr but these are
matters which the mathematical theory that considers only the value of
the pot and the probability of winning the hand has no concern with.
D&ring and Feger () see decision theory as a
normative theory for evaluating actions by considering expected
payoffs. Accordingly, they argue, &realism&, which
introduces practical considerations such as real-life limitations on
money supplies, or time, or whether anyone will really offer the game,
is irrelevant.
That sort of decision theory tells us that no amount is too great to
pay as an ideally rationally acceptable entrance fee for St.
Petersburg&a strange result, but one that does not force the
conclusion that there's something wrong with the theory. We might
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&File:Petersburg-Paradox.png& Wikimedia Commons,
18:12, 20 March 2006 version, original upload 30 September 2005.
(Colors were altered by the SEP editors.)
a preprint of a paper in Complexity, 5(4), 2000,
pp. 34&43, by Larry S. Liebovitch and Daniela Scheurle (Florida
Atlantic University).
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