I followed with interest the debate about Hamizah Nordin's choice of secondary school and the wider discussion about elite and neighborhood schools. One of the key areas of contention is whether elite schools merely possess an ostentatious ‘brand’ or truly provide a higher quality education justifying its higher costs.
Primarily because of the increasing investments in education across the developed world, this issue has in fact been at the center of much interesting economic research. These studies generally try to measure whether for instance a 30,000 USD per annum education at a Harvard or MIT produces correspondingly better economic returns (as measured by the increased future income of the student) than a 10,000 USD education at a more mundane college.
It may seem crass to reduce educational returns and ‘quality’ to merely increased future income. Yet as John Maynard Keynes, the great British economist, once put it: Economists are but the ‘keepers of the possibility of civilization’. Economic analysis deals with the quantifiable aspects of social phenomena--especially things with money signs attached, and provides a foundation for more refined analysis based on ethical, political and spiritual principles. When dealing with social issues, seemingly reductionist economic analysis is often indispensable for the possibility of clear thought.
The findings from these economic studies are in fact both surprising and instructive. Many early studies, beginning from the seminal 1963 research by Shane Hunt of Yale University, have shown that entering a college that is more selective in terms of SAT scores is positively and strongly correlated with higher future earnings. Thus it seems to be the case that attending an elite college does help a person to earn more in the future. To be exact, various studies have shown that an increase of 100 in average SAT admission score is correlated with a 3-7% increase in future earnings.
Yet a big question remains: are the increased earnings truly produced by a higher quality education at an elite school, or produced by a higher quality STUDENT? A variant of this point is the basic argument of many forum contributors. In fact Hunt (1963) did point out this ‘statistical bias’:
‘The C student from Princeton earns more than the A student from Podunk not mainly because of the prestige of a Princeton degree, but merely because he is abler. The golden touch is possessed not by an Ivy League college but by its students’
A Singaporean analogy will be how elite secondary schools take in students with incredible PSLE scores and then churn out equally incredible O or A Level results. The Midas touch of the school or simply golden boys and girls to begin with?
A number of recent studies have sought to address this problem, and the results should make egalitarian hearts leap. For instance, in an excellent Princeton University/Mellon Foundation econometric study by Krueger and Dale (1999), they selected for students with similar ‘quality’ but who attended different colleges. This is accomplished by matching pairs of students who have been accepted and rejected by the same sets of colleges, but who went to different colleges. Thus both student A and B may have been accepted by Harvard and SMU, but rejected by NUS, but one went to Harvard and the other to SMU. Then their future earnings are compared.
The findings from this ‘corrected’ study is striking. There is found to be little statistically significant difference in earnings between 2 students of similar ‘quality’ but who went to colleges of different selectivity. Thus for instance 519 matched students who went to moderately selective colleges (1000-1099 College Board scores) and highly selective colleges (1275 and above) earn much the same so long as their relative quality is similar. It appears to be golden students and not the Midas touch after all.
But before we get too carried away, there are 2 sobering caveats. The first is that future earnings may not be correlated with school selectivity, but it is positively correlated with school fees. It appears that the more expensive your college is, the higher your future earnings will be. Krueger and Dale are in fact unsure why this should be so. But one possibility is that higher fees do allow better facilities and more effective bidding for better professors. A more expensive education appears to be correlated with quality, or at least, student profitability.
How far is that relevant to Singapore schools? For one thing our fees are kept artificially low by subsidies, so school fees actually offer little guide to school expenditure and general investment in students. We might do better to look at costs than revenues. And the elite schools of Singapore do generally invest more in each student (in terms of per capita fixed and variable costs) than neighborhood schools, i.e. school selectivity and the expenses of education do run hand in hand. So far as this holds true, there is in fact a case that generous elite schools do provide a more profitable educational experience for their students than a neighborhood school.
The second caveat is linked to the first, and is directly relevant to the case of Hamizah Nordin. While for most students, there is no link between school selectivity and increased future earnings, Krueger and Dale found a strong positive link for low-income students. It appears that students from the bottom quarter of income levels do increase their future earnings significantly if they go to an elite school. This result is even stronger when it is both an elite AND expensive school.
This result corresponds to an argument made by many in support of elite schools: a truly meritocratic elite school system allows students from poor families to ascend to the higher rungs of society. The reasons for this are many. The first may be the more generous investment in education at elite schools. Other reasons may include the old boy-girl network that could be acquired and the contacts with richer friends that may elevate social aspirations. An intense atmosphere of competition and mutual emulation may also be contributing factors.
But for whatever reason, the economic analysis is clear. While we should not exaggerate the Midas touch of elite schools, yet so long as they aid students in terms of fees and benefit poor but able students like Hamizah Nordin, these institutions are in fact a help to both egalitarianism and social mobility. This is an important result that will perhaps help us to think more clearly about the true costs and benefits of elite schools. Before we dismiss them with a wave of the hand as fortresses for the snobbish and rich, we must remember that they are also havens of justice for the poor.
On a side note, schools like ACS with supposedly snobbish students can probably do more to shed their negative image by publicizing some equivalent of a ‘widows and orphans fund’ than by futile protests to shift subjective public opinion. And it will be in harmony with their Christian roots.
Saturday, December 17, 2005
Tuesday, December 06, 2005
Warcraft Economics
The following article will only be comprehensible to Warcraft III players:
Most players of Warcraft III will agree that the 'insane' level computer is a worthy opponent and that it takes a reasonable amount of patience and training to reach the skill level needed to be consistently victorious. Indeed, there are even rumors that the insane computer 'cheats'. This is perhaps because the computer does produce an unearthly number of units at a very distressing pace.
Yet, using some economic concepts, it is actually possible to work out a strategy that overwhelms and consistently annihilates an insane computer. This strategy works best against insane humans, night elves and less so against orcs (but you still can beat them quite easily). For reasons that will be clear to experienced warcraft players, it works least well against undead--but even then I have managed to use this strategy to beat them most of the time.
Before outlining the strategy, let us first consider the following:
Economic analysis of Warcraft
Efficiency
Economics studies how society extracts the highest 'value' from its limited resources. The greater the value produced from a given amount of resources, the higher is the 'efficiency' and productivity of an economy. An economically sound strategy in Warcraft will therefore be one that produces the greatest quality and quantity of military units from the limited resources of gold and lumber.
Can we measure this exactly? In one sense, yes. At the end of the game, there will be scores given for units and resources collected. The ratio between units and resources will be correlated to your efficiency. From my experience, the ratio of the [defeated] insane computer is typically 1:1. In other words, the computer has used 1 unit of resource to produce 1 unit of military force.
On the other hand, the strategy I use typically produces ratios of 2:1 or 3:1. If we factor in the hero score, the ratio can shoot much higher. In other words, the strategy outbuilds the computer by producing more and better units from less resources. It is no wonder that while I almost always collect less resources than the computer (I do not generally bother to expand), I still manage to field a force that is much more powerful than that of the computer.
Time
It is important to note that a more efficient strategy that produces more units from less resources, is also one that produces more units in less time. This is because the rate of gold collection is basically fixed and at its maximum, is roughly 20 gold/sec per base. In other words, in Warcraft, time is money, and how well you use your gold is really equivalent to how well you use your time. Lumber is not a truly limiting factor since its rate of collection can vary enormously dependent on your labor force, research level, faction etc.
Indeed, the strategy I shall outline below pits a goodly number of level 3 units against an insane computer with mostly level 1 and 2 units. This is achieved through the sheer speed of build-up.
Output value and opportunity costs
Given that efficiency and producitvity is related to output value/input value, how then do we measure output value? While the game puts an approximate numerical value to our units at the end of the game, yet is it possible to have a more complete measurement of 'military value'?
Actually it is quite difficult to have any exact measurement. Military value is determined by its hit points, damage, range, armor, special abilities and whether it is a flying, siege, melee or range unit. In different battles, different aspects of the unit may be helpful or become fatal. Thus, the military value of units differ from battle to battle. A tauren may be worth a lot fighting against footmen, but is utterly worthless against a frost wyrm. A human siege engine is very useful if you are attacking a town, but generally worthless otherwise.
Thus we seem to be stuck with a mostly approximate measurement of efficiency, because output value is so tentative. Yet economics always insists on 'opportunity costs': that due to limited resources, if you do something, you have to give up the next best alternative. Thus if you build a crypt fiend, you have to forgo 2 ghouls. If you go for the chimaera, you may have to give up your mountain giant. It is therefore supremely important to know which are the 'best units' to build to maximise your output value and efficiency.
Battle efficiency
This brings us to another important point. The key to winning warcraft is not just productive efficiency (producing more/better units from less resources) but battle efficiency. What do I mean? Basically in any battle, we aim to bring an army with a far higher [approximate] military value against a computer with less (or even better, non-existent) military value. It is indeed possible to lose a game by having high productive efficiency but low battle efficiency.
For instance (to use an extreme example) you might foolishly but efficiently build up 6 taurens and a tauren chieftain and send them against your inefficient undead foe with only level 2 units. Yet just ONE gargoyle will suffice to annihilate (albeit very slowly) your whole army. Or you may build 10 towers and ring them around your base. Yet just one siege unit is enough to destroy everything.
I have found that the best way to ensure high battle efficiency is to use flying units. The reason is obvious: flying units are vulnerable only to other flying units and range units. Typically the computer will only have a relatively small number of these and you can easily achieve overwhelming battle efficiency with enough flyers. Of course, crypt fiends and (to a lesser extent) raiders can reduce the efficiency by sending our units to the ground. Towers can also alter the balance.
But even with these caveats, the computer generally produce far too few crypt fiends, raiders (the computer take a long time to get to the ensnare ability), range or flying units. Thus my strategy will be one that uses a swarm of flying units to translate a high productive efficiency into an overwhelming battle efficiency.
Warcraft investment
Like in the real world economy, Warcraft productivity is achieved by investments. Basically the early choices of Warcraft are determined by whether we want to 'consume' the resources in building poor level 1 units, or 'invest' in upgrades that will allow us to build better units at the same cost, i.e. become more efficient.
Simple calculations will tell you that higher efficiency can only be achieved if we consume a minimum of resources in the early stages (just enough to defend your base), invest as much as we can, and save the rest in order to build the best units. Thus we try to 'tech', but safely.
The Strategy
This strategy works best if you take humans. The reasons should become apparent as I describe it.
Basically, we seek to get the highest value out of each unit of resource used and achieve the highest productive and battle efficiency. The way I have found to be most effective against the insane computer is to build a swarm of gryphon riders, dragonhawk riders and one paladin.
First, we settle the question of resource collection. Basically I find that 5 peasants collecting gold, 7 to the trees, and 2 building at all times, is quite an optimum arrangement. At the beginning, I will send 3 peasants to the mine and 2 to build a farm and an altar. The other 9 peasants will be produced and sent on their respective ways.
Now as stated above, we must defend our base by consuming a minimum of resources. How can this be done? The first way is to simply depend on your 14 peasant militia and one hero. While the 14 militia can beat off most early attacks, this policy involves undue risk. You can lose a battle and your whole base before you build your first gryphon rider. And you are likely to lose peasants in any attack. And most seriously, you slow down your resource collection each time the computer comes calling. This hardly constitutes productive efficiency.
A much better way (and one chief reason for using humans) is to use the 14 militia as a final backup. My way is to build 3 adjoined towers surrounded by farms. Why? I think it is hard to deny that towers using focused fire are indeed the most cost effective DEFENSIVE units in the early game. NO early attack (meaning one without siege units) from the insane computer can destroy a base with 3 towers, and if needed, 14 militia and one hero. Indeed, you don't have to call on your peasants at all. Best of all, 3 towers are really quite cheap and does not consume food.
So far, my worst experience is when an orc army came down and blew up 2 towers. But that was far too late, for 2 gryphon riders were produced during the battle and quickly routed them. I did not call on a single peasant.
So how to go about the tower construction? First I usually get 1 peasant to build 2 farms while another produces an altar. Once the altar is up, that peasant will build up a lumber mill (necessary to produce the tower upgrade) while the farm peasant start building 3 towers. When the mill is done, we upgrade the 3 scout towers to defensive ones. Meanwhile I will also research one level of masonry upgrade at the mill for 'kiasu' purposes. The 2 peasants will then continue building 4 more farms (6 altogether) to both ring the towers and to provide the food for the upcoming air units.
Upgrading to reach gryphon riders
With the defence settled, we now deal with the upgrading process. Basically in 'teching', there is only one rule: secure your base and then upgrade to castle above all else. Once my 9 peasants are out, I immediately upgrade (the resources are just enough at that point usually), then once stage 2 is done, I immediately start the castle upgrade. You must plan your resource usage for farms, towers and lumber collection upgrades (I usually upgrade once) in such a way that you are not even the least delayed in your upgrades. With this strategy you can reach Castle while the insane computer are still churning out mostly level 1 units.
Besides the upgrades you have the resources to build the following: At stage 2, after finishing 6 farms, you build an arcane vault, then start on 2 Gryphon aviaries. Usually I will have 2 aviaries and have produced one dragonhawk rider before Castle rolls along.
At this point, you would have saved lots of gold and lumber. Thus use your savings to produce gryphon riders. You would have enough to produce about 4 very quickly before being forced to slow down. Meanwhile one peasant will continue building farms (you can intially slow down on farm production since you already have a surplus), while another will start on the blacksmith and then barracks (for the upgrades).
Generally, even against the undead, you can feel quite safe when you reach about 8 gryphon riders and 2 dragonhawk riders. Obviously you should have started creeping with less and will have a decent level 3-4 paladin by now. The paladin should focus on divine shield (since he is alone on the ground--this ability is the main reason why the paladin is chosen) and devotion aura. He don't need to learn resurrection since flying units die without corpses.
With such an army (and more gryphon riders on the way), you can start creeping around the enemy base. For non-undead races, you can even attack the base directly (once you have the Cloud ability for dragonhawk riders) and finish the game speedily. The undead base is apt to have an unseemly number of Towers, necromancers, crypt fiends and other irritants. It is extremely unwise to attack them directly. You really would not want to cloud their 4 towers and then have your hawk rider webbed or sent to sleep...
For any faction, I usually prefer to lure the insane computer out and crush their huge but mainly worthless and primitive army.
There are several ways to lure them out. One expensive but infallible way is to build an expansion town. The truly insane computer will quickly dispatch a suicide mission towards your expansion, and you can wait for them at your leisure while increasing your air force. The outcome cannot be more certain. I usually prefer to attack THEIR expansion. That will also send their whole army out in a hurry.
Against non-undead computer players, victory is relatively straightforward. For humans, exterminate their few riflemen and the rest will be gone. Due to your production speed, they will have no gryphon riders or hawks (and even if they do, you have your aerial shackles and overwhelming numbers). For night elves, use storm hammer upgrades to exterminate their archers. Dryads (which are immune to gryphon magical attacks) are a special problem, but your hawk riders and paladin can deal with them. For orcs, kill their trolls and their raiders. Their raiders (if any) typically cannot ensnare yet. Orc towers and burrows are very fragile and indeed you don't even need dragonhawk riders. Just build up a mere 6 gryphons and usually it is enough to wipe out the orc army and demolish an entire base (though 8 will be more efficient).
In general, against non undead factions, your overwhelming air superiority, cloud/aerial shackles and your invincible (temporarily) paladin will allow you to exterminate them easily. Typically I demolish their expansion, send their defending army fleeing, and then harass them endlessly (lure them out, crush them etc.) till they are no more. The paladin typically reach level 9-10 at the end of it. Best try to secure some Fountain of health near their base for maximum ease of attack.
For a full undead army with crypt fiends, gargoyles, a dread lord with sleep and even frost wyrms, things can get nasty. But even then, aerial shackles can deal with the frost wyrm[s] (they are indeed the least of my concerns), and sufficient numbers (around 8) can make short work of the gargolyes even with some gryphons being webbed. Of course numbers will allow you to kill the crypt fiends quickly (you MUST). Still casualties can be significant, and you can even lose your first battle.
However so long as you don't hit the main base but lure them out, the insane computer tend not to concentrate its forces, but disperse them randomly, first sending gargoyles and frost wyrms to their doom, before the crypt fiends reach the scene. So most of the time the strategy still works. Of course you can simply build 15 gryphon riders etc., but the longer you wait the stronger the undead opponent becomes.
Such is the natural disadvantage of using air units against undead. I have been thinking of some superior and economically sound strategy, but no fruits yet. Any suggestions?
Hope this will provide some fruit for thoughts. For those who are interested, you can try it out and tell me what you think. The skill level needed to win with this strategy is quite modest. By the way, for obvious reasons, this strategy cannot work against non-computer opponents who knows you are going to use a swarm of air units. This is a strategy designed to beat predictably insane computers.
Most players of Warcraft III will agree that the 'insane' level computer is a worthy opponent and that it takes a reasonable amount of patience and training to reach the skill level needed to be consistently victorious. Indeed, there are even rumors that the insane computer 'cheats'. This is perhaps because the computer does produce an unearthly number of units at a very distressing pace.
Yet, using some economic concepts, it is actually possible to work out a strategy that overwhelms and consistently annihilates an insane computer. This strategy works best against insane humans, night elves and less so against orcs (but you still can beat them quite easily). For reasons that will be clear to experienced warcraft players, it works least well against undead--but even then I have managed to use this strategy to beat them most of the time.
Before outlining the strategy, let us first consider the following:
Economic analysis of Warcraft
Efficiency
Economics studies how society extracts the highest 'value' from its limited resources. The greater the value produced from a given amount of resources, the higher is the 'efficiency' and productivity of an economy. An economically sound strategy in Warcraft will therefore be one that produces the greatest quality and quantity of military units from the limited resources of gold and lumber.
Can we measure this exactly? In one sense, yes. At the end of the game, there will be scores given for units and resources collected. The ratio between units and resources will be correlated to your efficiency. From my experience, the ratio of the [defeated] insane computer is typically 1:1. In other words, the computer has used 1 unit of resource to produce 1 unit of military force.
On the other hand, the strategy I use typically produces ratios of 2:1 or 3:1. If we factor in the hero score, the ratio can shoot much higher. In other words, the strategy outbuilds the computer by producing more and better units from less resources. It is no wonder that while I almost always collect less resources than the computer (I do not generally bother to expand), I still manage to field a force that is much more powerful than that of the computer.
Time
It is important to note that a more efficient strategy that produces more units from less resources, is also one that produces more units in less time. This is because the rate of gold collection is basically fixed and at its maximum, is roughly 20 gold/sec per base. In other words, in Warcraft, time is money, and how well you use your gold is really equivalent to how well you use your time. Lumber is not a truly limiting factor since its rate of collection can vary enormously dependent on your labor force, research level, faction etc.
Indeed, the strategy I shall outline below pits a goodly number of level 3 units against an insane computer with mostly level 1 and 2 units. This is achieved through the sheer speed of build-up.
Output value and opportunity costs
Given that efficiency and producitvity is related to output value/input value, how then do we measure output value? While the game puts an approximate numerical value to our units at the end of the game, yet is it possible to have a more complete measurement of 'military value'?
Actually it is quite difficult to have any exact measurement. Military value is determined by its hit points, damage, range, armor, special abilities and whether it is a flying, siege, melee or range unit. In different battles, different aspects of the unit may be helpful or become fatal. Thus, the military value of units differ from battle to battle. A tauren may be worth a lot fighting against footmen, but is utterly worthless against a frost wyrm. A human siege engine is very useful if you are attacking a town, but generally worthless otherwise.
Thus we seem to be stuck with a mostly approximate measurement of efficiency, because output value is so tentative. Yet economics always insists on 'opportunity costs': that due to limited resources, if you do something, you have to give up the next best alternative. Thus if you build a crypt fiend, you have to forgo 2 ghouls. If you go for the chimaera, you may have to give up your mountain giant. It is therefore supremely important to know which are the 'best units' to build to maximise your output value and efficiency.
Battle efficiency
This brings us to another important point. The key to winning warcraft is not just productive efficiency (producing more/better units from less resources) but battle efficiency. What do I mean? Basically in any battle, we aim to bring an army with a far higher [approximate] military value against a computer with less (or even better, non-existent) military value. It is indeed possible to lose a game by having high productive efficiency but low battle efficiency.
For instance (to use an extreme example) you might foolishly but efficiently build up 6 taurens and a tauren chieftain and send them against your inefficient undead foe with only level 2 units. Yet just ONE gargoyle will suffice to annihilate (albeit very slowly) your whole army. Or you may build 10 towers and ring them around your base. Yet just one siege unit is enough to destroy everything.
I have found that the best way to ensure high battle efficiency is to use flying units. The reason is obvious: flying units are vulnerable only to other flying units and range units. Typically the computer will only have a relatively small number of these and you can easily achieve overwhelming battle efficiency with enough flyers. Of course, crypt fiends and (to a lesser extent) raiders can reduce the efficiency by sending our units to the ground. Towers can also alter the balance.
But even with these caveats, the computer generally produce far too few crypt fiends, raiders (the computer take a long time to get to the ensnare ability), range or flying units. Thus my strategy will be one that uses a swarm of flying units to translate a high productive efficiency into an overwhelming battle efficiency.
Warcraft investment
Like in the real world economy, Warcraft productivity is achieved by investments. Basically the early choices of Warcraft are determined by whether we want to 'consume' the resources in building poor level 1 units, or 'invest' in upgrades that will allow us to build better units at the same cost, i.e. become more efficient.
Simple calculations will tell you that higher efficiency can only be achieved if we consume a minimum of resources in the early stages (just enough to defend your base), invest as much as we can, and save the rest in order to build the best units. Thus we try to 'tech', but safely.
The Strategy
This strategy works best if you take humans. The reasons should become apparent as I describe it.
Basically, we seek to get the highest value out of each unit of resource used and achieve the highest productive and battle efficiency. The way I have found to be most effective against the insane computer is to build a swarm of gryphon riders, dragonhawk riders and one paladin.
First, we settle the question of resource collection. Basically I find that 5 peasants collecting gold, 7 to the trees, and 2 building at all times, is quite an optimum arrangement. At the beginning, I will send 3 peasants to the mine and 2 to build a farm and an altar. The other 9 peasants will be produced and sent on their respective ways.
Now as stated above, we must defend our base by consuming a minimum of resources. How can this be done? The first way is to simply depend on your 14 peasant militia and one hero. While the 14 militia can beat off most early attacks, this policy involves undue risk. You can lose a battle and your whole base before you build your first gryphon rider. And you are likely to lose peasants in any attack. And most seriously, you slow down your resource collection each time the computer comes calling. This hardly constitutes productive efficiency.
A much better way (and one chief reason for using humans) is to use the 14 militia as a final backup. My way is to build 3 adjoined towers surrounded by farms. Why? I think it is hard to deny that towers using focused fire are indeed the most cost effective DEFENSIVE units in the early game. NO early attack (meaning one without siege units) from the insane computer can destroy a base with 3 towers, and if needed, 14 militia and one hero. Indeed, you don't have to call on your peasants at all. Best of all, 3 towers are really quite cheap and does not consume food.
So far, my worst experience is when an orc army came down and blew up 2 towers. But that was far too late, for 2 gryphon riders were produced during the battle and quickly routed them. I did not call on a single peasant.
So how to go about the tower construction? First I usually get 1 peasant to build 2 farms while another produces an altar. Once the altar is up, that peasant will build up a lumber mill (necessary to produce the tower upgrade) while the farm peasant start building 3 towers. When the mill is done, we upgrade the 3 scout towers to defensive ones. Meanwhile I will also research one level of masonry upgrade at the mill for 'kiasu' purposes. The 2 peasants will then continue building 4 more farms (6 altogether) to both ring the towers and to provide the food for the upcoming air units.
Upgrading to reach gryphon riders
With the defence settled, we now deal with the upgrading process. Basically in 'teching', there is only one rule: secure your base and then upgrade to castle above all else. Once my 9 peasants are out, I immediately upgrade (the resources are just enough at that point usually), then once stage 2 is done, I immediately start the castle upgrade. You must plan your resource usage for farms, towers and lumber collection upgrades (I usually upgrade once) in such a way that you are not even the least delayed in your upgrades. With this strategy you can reach Castle while the insane computer are still churning out mostly level 1 units.
Besides the upgrades you have the resources to build the following: At stage 2, after finishing 6 farms, you build an arcane vault, then start on 2 Gryphon aviaries. Usually I will have 2 aviaries and have produced one dragonhawk rider before Castle rolls along.
At this point, you would have saved lots of gold and lumber. Thus use your savings to produce gryphon riders. You would have enough to produce about 4 very quickly before being forced to slow down. Meanwhile one peasant will continue building farms (you can intially slow down on farm production since you already have a surplus), while another will start on the blacksmith and then barracks (for the upgrades).
Generally, even against the undead, you can feel quite safe when you reach about 8 gryphon riders and 2 dragonhawk riders. Obviously you should have started creeping with less and will have a decent level 3-4 paladin by now. The paladin should focus on divine shield (since he is alone on the ground--this ability is the main reason why the paladin is chosen) and devotion aura. He don't need to learn resurrection since flying units die without corpses.
With such an army (and more gryphon riders on the way), you can start creeping around the enemy base. For non-undead races, you can even attack the base directly (once you have the Cloud ability for dragonhawk riders) and finish the game speedily. The undead base is apt to have an unseemly number of Towers, necromancers, crypt fiends and other irritants. It is extremely unwise to attack them directly. You really would not want to cloud their 4 towers and then have your hawk rider webbed or sent to sleep...
For any faction, I usually prefer to lure the insane computer out and crush their huge but mainly worthless and primitive army.
There are several ways to lure them out. One expensive but infallible way is to build an expansion town. The truly insane computer will quickly dispatch a suicide mission towards your expansion, and you can wait for them at your leisure while increasing your air force. The outcome cannot be more certain. I usually prefer to attack THEIR expansion. That will also send their whole army out in a hurry.
Against non-undead computer players, victory is relatively straightforward. For humans, exterminate their few riflemen and the rest will be gone. Due to your production speed, they will have no gryphon riders or hawks (and even if they do, you have your aerial shackles and overwhelming numbers). For night elves, use storm hammer upgrades to exterminate their archers. Dryads (which are immune to gryphon magical attacks) are a special problem, but your hawk riders and paladin can deal with them. For orcs, kill their trolls and their raiders. Their raiders (if any) typically cannot ensnare yet. Orc towers and burrows are very fragile and indeed you don't even need dragonhawk riders. Just build up a mere 6 gryphons and usually it is enough to wipe out the orc army and demolish an entire base (though 8 will be more efficient).
In general, against non undead factions, your overwhelming air superiority, cloud/aerial shackles and your invincible (temporarily) paladin will allow you to exterminate them easily. Typically I demolish their expansion, send their defending army fleeing, and then harass them endlessly (lure them out, crush them etc.) till they are no more. The paladin typically reach level 9-10 at the end of it. Best try to secure some Fountain of health near their base for maximum ease of attack.
For a full undead army with crypt fiends, gargoyles, a dread lord with sleep and even frost wyrms, things can get nasty. But even then, aerial shackles can deal with the frost wyrm[s] (they are indeed the least of my concerns), and sufficient numbers (around 8) can make short work of the gargolyes even with some gryphons being webbed. Of course numbers will allow you to kill the crypt fiends quickly (you MUST). Still casualties can be significant, and you can even lose your first battle.
However so long as you don't hit the main base but lure them out, the insane computer tend not to concentrate its forces, but disperse them randomly, first sending gargoyles and frost wyrms to their doom, before the crypt fiends reach the scene. So most of the time the strategy still works. Of course you can simply build 15 gryphon riders etc., but the longer you wait the stronger the undead opponent becomes.
Such is the natural disadvantage of using air units against undead. I have been thinking of some superior and economically sound strategy, but no fruits yet. Any suggestions?
Hope this will provide some fruit for thoughts. For those who are interested, you can try it out and tell me what you think. The skill level needed to win with this strategy is quite modest. By the way, for obvious reasons, this strategy cannot work against non-computer opponents who knows you are going to use a swarm of air units. This is a strategy designed to beat predictably insane computers.
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